Polytope of Type {816}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {816}*1632
Also Known As : 816-gon, {816}. if this polytope has another name.
Group : SmallGroup(1632,160)
Rank : 2
Schlafli Type : {816}
Number of vertices, edges, etc : 816, 816
Order of s₀s₁ : 816
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {408}*816
   3-fold quotients : {272}*544
   4-fold quotients : {204}*408
   6-fold quotients : {136}*272
   8-fold quotients : {102}*204
   12-fold quotients : {68}*136
   16-fold quotients : {51}*102
   17-fold quotients : {48}*96
   24-fold quotients : {34}*68
   34-fold quotients : {24}*48
   48-fold quotients : {17}*34
   51-fold quotients : {16}*32
   68-fold quotients : {12}*24
   102-fold quotients : {8}*16
   136-fold quotients : {6}*12
   204-fold quotients : {4}*8
   272-fold quotients : {3}*6
   408-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)
( 18, 35)( 19, 51)( 20, 50)( 21, 49)( 22, 48)( 23, 47)( 24, 46)( 25, 45)
( 26, 44)( 27, 43)( 28, 42)( 29, 41)( 30, 40)( 31, 39)( 32, 38)( 33, 37)
( 34, 36)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)
( 60, 61)( 69, 86)( 70,102)( 71,101)( 72,100)( 73, 99)( 74, 98)( 75, 97)
( 76, 96)( 77, 95)( 78, 94)( 79, 93)( 80, 92)( 81, 91)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)(103,154)(104,170)(105,169)(106,168)(107,167)(108,166)
(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)
(117,157)(118,156)(119,155)(120,188)(121,204)(122,203)(123,202)(124,201)
(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)(132,193)
(133,192)(134,191)(135,190)(136,189)(137,171)(138,187)(139,186)(140,185)
(141,184)(142,183)(143,182)(144,181)(145,180)(146,179)(147,178)(148,177)
(149,176)(150,175)(151,174)(152,173)(153,172)(205,307)(206,323)(207,322)
(208,321)(209,320)(210,319)(211,318)(212,317)(213,316)(214,315)(215,314)
(216,313)(217,312)(218,311)(219,310)(220,309)(221,308)(222,341)(223,357)
(224,356)(225,355)(226,354)(227,353)(228,352)(229,351)(230,350)(231,349)
(232,348)(233,347)(234,346)(235,345)(236,344)(237,343)(238,342)(239,324)
(240,340)(241,339)(242,338)(243,337)(244,336)(245,335)(246,334)(247,333)
(248,332)(249,331)(250,330)(251,329)(252,328)(253,327)(254,326)(255,325)
(256,358)(257,374)(258,373)(259,372)(260,371)(261,370)(262,369)(263,368)
(264,367)(265,366)(266,365)(267,364)(268,363)(269,362)(270,361)(271,360)
(272,359)(273,392)(274,408)(275,407)(276,406)(277,405)(278,404)(279,403)
(280,402)(281,401)(282,400)(283,399)(284,398)(285,397)(286,396)(287,395)
(288,394)(289,393)(290,375)(291,391)(292,390)(293,389)(294,388)(295,387)
(296,386)(297,385)(298,384)(299,383)(300,382)(301,381)(302,380)(303,379)
(304,378)(305,377)(306,376)(409,613)(410,629)(411,628)(412,627)(413,626)
(414,625)(415,624)(416,623)(417,622)(418,621)(419,620)(420,619)(421,618)
(422,617)(423,616)(424,615)(425,614)(426,647)(427,663)(428,662)(429,661)
(430,660)(431,659)(432,658)(433,657)(434,656)(435,655)(436,654)(437,653)
(438,652)(439,651)(440,650)(441,649)(442,648)(443,630)(444,646)(445,645)
(446,644)(447,643)(448,642)(449,641)(450,640)(451,639)(452,638)(453,637)
(454,636)(455,635)(456,634)(457,633)(458,632)(459,631)(460,664)(461,680)
(462,679)(463,678)(464,677)(465,676)(466,675)(467,674)(468,673)(469,672)
(470,671)(471,670)(472,669)(473,668)(474,667)(475,666)(476,665)(477,698)
(478,714)(479,713)(480,712)(481,711)(482,710)(483,709)(484,708)(485,707)
(486,706)(487,705)(488,704)(489,703)(490,702)(491,701)(492,700)(493,699)
(494,681)(495,697)(496,696)(497,695)(498,694)(499,693)(500,692)(501,691)
(502,690)(503,689)(504,688)(505,687)(506,686)(507,685)(508,684)(509,683)
(510,682)(511,766)(512,782)(513,781)(514,780)(515,779)(516,778)(517,777)
(518,776)(519,775)(520,774)(521,773)(522,772)(523,771)(524,770)(525,769)
(526,768)(527,767)(528,800)(529,816)(530,815)(531,814)(532,813)(533,812)
(534,811)(535,810)(536,809)(537,808)(538,807)(539,806)(540,805)(541,804)
(542,803)(543,802)(544,801)(545,783)(546,799)(547,798)(548,797)(549,796)
(550,795)(551,794)(552,793)(553,792)(554,791)(555,790)(556,789)(557,788)
(558,787)(559,786)(560,785)(561,784)(562,715)(563,731)(564,730)(565,729)
(566,728)(567,727)(568,726)(569,725)(570,724)(571,723)(572,722)(573,721)
(574,720)(575,719)(576,718)(577,717)(578,716)(579,749)(580,765)(581,764)
(582,763)(583,762)(584,761)(585,760)(586,759)(587,758)(588,757)(589,756)
(590,755)(591,754)(592,753)(593,752)(594,751)(595,750)(596,732)(597,748)
(598,747)(599,746)(600,745)(601,744)(602,743)(603,742)(604,741)(605,740)
(606,739)(607,738)(608,737)(609,736)(610,735)(611,734)(612,733);;
s1 := (  1,427)(  2,426)(  3,442)(  4,441)(  5,440)(  6,439)(  7,438)(  8,437)
(  9,436)( 10,435)( 11,434)( 12,433)( 13,432)( 14,431)( 15,430)( 16,429)
( 17,428)( 18,410)( 19,409)( 20,425)( 21,424)( 22,423)( 23,422)( 24,421)
( 25,420)( 26,419)( 27,418)( 28,417)( 29,416)( 30,415)( 31,414)( 32,413)
( 33,412)( 34,411)( 35,444)( 36,443)( 37,459)( 38,458)( 39,457)( 40,456)
( 41,455)( 42,454)( 43,453)( 44,452)( 45,451)( 46,450)( 47,449)( 48,448)
( 49,447)( 50,446)( 51,445)( 52,478)( 53,477)( 54,493)( 55,492)( 56,491)
( 57,490)( 58,489)( 59,488)( 60,487)( 61,486)( 62,485)( 63,484)( 64,483)
( 65,482)( 66,481)( 67,480)( 68,479)( 69,461)( 70,460)( 71,476)( 72,475)
( 73,474)( 74,473)( 75,472)( 76,471)( 77,470)( 78,469)( 79,468)( 80,467)
( 81,466)( 82,465)( 83,464)( 84,463)( 85,462)( 86,495)( 87,494)( 88,510)
( 89,509)( 90,508)( 91,507)( 92,506)( 93,505)( 94,504)( 95,503)( 96,502)
( 97,501)( 98,500)( 99,499)(100,498)(101,497)(102,496)(103,580)(104,579)
(105,595)(106,594)(107,593)(108,592)(109,591)(110,590)(111,589)(112,588)
(113,587)(114,586)(115,585)(116,584)(117,583)(118,582)(119,581)(120,563)
(121,562)(122,578)(123,577)(124,576)(125,575)(126,574)(127,573)(128,572)
(129,571)(130,570)(131,569)(132,568)(133,567)(134,566)(135,565)(136,564)
(137,597)(138,596)(139,612)(140,611)(141,610)(142,609)(143,608)(144,607)
(145,606)(146,605)(147,604)(148,603)(149,602)(150,601)(151,600)(152,599)
(153,598)(154,529)(155,528)(156,544)(157,543)(158,542)(159,541)(160,540)
(161,539)(162,538)(163,537)(164,536)(165,535)(166,534)(167,533)(168,532)
(169,531)(170,530)(171,512)(172,511)(173,527)(174,526)(175,525)(176,524)
(177,523)(178,522)(179,521)(180,520)(181,519)(182,518)(183,517)(184,516)
(185,515)(186,514)(187,513)(188,546)(189,545)(190,561)(191,560)(192,559)
(193,558)(194,557)(195,556)(196,555)(197,554)(198,553)(199,552)(200,551)
(201,550)(202,549)(203,548)(204,547)(205,733)(206,732)(207,748)(208,747)
(209,746)(210,745)(211,744)(212,743)(213,742)(214,741)(215,740)(216,739)
(217,738)(218,737)(219,736)(220,735)(221,734)(222,716)(223,715)(224,731)
(225,730)(226,729)(227,728)(228,727)(229,726)(230,725)(231,724)(232,723)
(233,722)(234,721)(235,720)(236,719)(237,718)(238,717)(239,750)(240,749)
(241,765)(242,764)(243,763)(244,762)(245,761)(246,760)(247,759)(248,758)
(249,757)(250,756)(251,755)(252,754)(253,753)(254,752)(255,751)(256,784)
(257,783)(258,799)(259,798)(260,797)(261,796)(262,795)(263,794)(264,793)
(265,792)(266,791)(267,790)(268,789)(269,788)(270,787)(271,786)(272,785)
(273,767)(274,766)(275,782)(276,781)(277,780)(278,779)(279,778)(280,777)
(281,776)(282,775)(283,774)(284,773)(285,772)(286,771)(287,770)(288,769)
(289,768)(290,801)(291,800)(292,816)(293,815)(294,814)(295,813)(296,812)
(297,811)(298,810)(299,809)(300,808)(301,807)(302,806)(303,805)(304,804)
(305,803)(306,802)(307,631)(308,630)(309,646)(310,645)(311,644)(312,643)
(313,642)(314,641)(315,640)(316,639)(317,638)(318,637)(319,636)(320,635)
(321,634)(322,633)(323,632)(324,614)(325,613)(326,629)(327,628)(328,627)
(329,626)(330,625)(331,624)(332,623)(333,622)(334,621)(335,620)(336,619)
(337,618)(338,617)(339,616)(340,615)(341,648)(342,647)(343,663)(344,662)
(345,661)(346,660)(347,659)(348,658)(349,657)(350,656)(351,655)(352,654)
(353,653)(354,652)(355,651)(356,650)(357,649)(358,682)(359,681)(360,697)
(361,696)(362,695)(363,694)(364,693)(365,692)(366,691)(367,690)(368,689)
(369,688)(370,687)(371,686)(372,685)(373,684)(374,683)(375,665)(376,664)
(377,680)(378,679)(379,678)(380,677)(381,676)(382,675)(383,674)(384,673)
(385,672)(386,671)(387,670)(388,669)(389,668)(390,667)(391,666)(392,699)
(393,698)(394,714)(395,713)(396,712)(397,711)(398,710)(399,709)(400,708)
(401,707)(402,706)(403,705)(404,704)(405,703)(406,702)(407,701)(408,700);;
poly := Group([s0,s1]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(816)!(  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)
(  9, 10)( 18, 35)( 19, 51)( 20, 50)( 21, 49)( 22, 48)( 23, 47)( 24, 46)
( 25, 45)( 26, 44)( 27, 43)( 28, 42)( 29, 41)( 30, 40)( 31, 39)( 32, 38)
( 33, 37)( 34, 36)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)
( 59, 62)( 60, 61)( 69, 86)( 70,102)( 71,101)( 72,100)( 73, 99)( 74, 98)
( 75, 97)( 76, 96)( 77, 95)( 78, 94)( 79, 93)( 80, 92)( 81, 91)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)(103,154)(104,170)(105,169)(106,168)(107,167)
(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)
(116,158)(117,157)(118,156)(119,155)(120,188)(121,204)(122,203)(123,202)
(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)
(132,193)(133,192)(134,191)(135,190)(136,189)(137,171)(138,187)(139,186)
(140,185)(141,184)(142,183)(143,182)(144,181)(145,180)(146,179)(147,178)
(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(205,307)(206,323)
(207,322)(208,321)(209,320)(210,319)(211,318)(212,317)(213,316)(214,315)
(215,314)(216,313)(217,312)(218,311)(219,310)(220,309)(221,308)(222,341)
(223,357)(224,356)(225,355)(226,354)(227,353)(228,352)(229,351)(230,350)
(231,349)(232,348)(233,347)(234,346)(235,345)(236,344)(237,343)(238,342)
(239,324)(240,340)(241,339)(242,338)(243,337)(244,336)(245,335)(246,334)
(247,333)(248,332)(249,331)(250,330)(251,329)(252,328)(253,327)(254,326)
(255,325)(256,358)(257,374)(258,373)(259,372)(260,371)(261,370)(262,369)
(263,368)(264,367)(265,366)(266,365)(267,364)(268,363)(269,362)(270,361)
(271,360)(272,359)(273,392)(274,408)(275,407)(276,406)(277,405)(278,404)
(279,403)(280,402)(281,401)(282,400)(283,399)(284,398)(285,397)(286,396)
(287,395)(288,394)(289,393)(290,375)(291,391)(292,390)(293,389)(294,388)
(295,387)(296,386)(297,385)(298,384)(299,383)(300,382)(301,381)(302,380)
(303,379)(304,378)(305,377)(306,376)(409,613)(410,629)(411,628)(412,627)
(413,626)(414,625)(415,624)(416,623)(417,622)(418,621)(419,620)(420,619)
(421,618)(422,617)(423,616)(424,615)(425,614)(426,647)(427,663)(428,662)
(429,661)(430,660)(431,659)(432,658)(433,657)(434,656)(435,655)(436,654)
(437,653)(438,652)(439,651)(440,650)(441,649)(442,648)(443,630)(444,646)
(445,645)(446,644)(447,643)(448,642)(449,641)(450,640)(451,639)(452,638)
(453,637)(454,636)(455,635)(456,634)(457,633)(458,632)(459,631)(460,664)
(461,680)(462,679)(463,678)(464,677)(465,676)(466,675)(467,674)(468,673)
(469,672)(470,671)(471,670)(472,669)(473,668)(474,667)(475,666)(476,665)
(477,698)(478,714)(479,713)(480,712)(481,711)(482,710)(483,709)(484,708)
(485,707)(486,706)(487,705)(488,704)(489,703)(490,702)(491,701)(492,700)
(493,699)(494,681)(495,697)(496,696)(497,695)(498,694)(499,693)(500,692)
(501,691)(502,690)(503,689)(504,688)(505,687)(506,686)(507,685)(508,684)
(509,683)(510,682)(511,766)(512,782)(513,781)(514,780)(515,779)(516,778)
(517,777)(518,776)(519,775)(520,774)(521,773)(522,772)(523,771)(524,770)
(525,769)(526,768)(527,767)(528,800)(529,816)(530,815)(531,814)(532,813)
(533,812)(534,811)(535,810)(536,809)(537,808)(538,807)(539,806)(540,805)
(541,804)(542,803)(543,802)(544,801)(545,783)(546,799)(547,798)(548,797)
(549,796)(550,795)(551,794)(552,793)(553,792)(554,791)(555,790)(556,789)
(557,788)(558,787)(559,786)(560,785)(561,784)(562,715)(563,731)(564,730)
(565,729)(566,728)(567,727)(568,726)(569,725)(570,724)(571,723)(572,722)
(573,721)(574,720)(575,719)(576,718)(577,717)(578,716)(579,749)(580,765)
(581,764)(582,763)(583,762)(584,761)(585,760)(586,759)(587,758)(588,757)
(589,756)(590,755)(591,754)(592,753)(593,752)(594,751)(595,750)(596,732)
(597,748)(598,747)(599,746)(600,745)(601,744)(602,743)(603,742)(604,741)
(605,740)(606,739)(607,738)(608,737)(609,736)(610,735)(611,734)(612,733);
s1 := Sym(816)!(  1,427)(  2,426)(  3,442)(  4,441)(  5,440)(  6,439)(  7,438)
(  8,437)(  9,436)( 10,435)( 11,434)( 12,433)( 13,432)( 14,431)( 15,430)
( 16,429)( 17,428)( 18,410)( 19,409)( 20,425)( 21,424)( 22,423)( 23,422)
( 24,421)( 25,420)( 26,419)( 27,418)( 28,417)( 29,416)( 30,415)( 31,414)
( 32,413)( 33,412)( 34,411)( 35,444)( 36,443)( 37,459)( 38,458)( 39,457)
( 40,456)( 41,455)( 42,454)( 43,453)( 44,452)( 45,451)( 46,450)( 47,449)
( 48,448)( 49,447)( 50,446)( 51,445)( 52,478)( 53,477)( 54,493)( 55,492)
( 56,491)( 57,490)( 58,489)( 59,488)( 60,487)( 61,486)( 62,485)( 63,484)
( 64,483)( 65,482)( 66,481)( 67,480)( 68,479)( 69,461)( 70,460)( 71,476)
( 72,475)( 73,474)( 74,473)( 75,472)( 76,471)( 77,470)( 78,469)( 79,468)
( 80,467)( 81,466)( 82,465)( 83,464)( 84,463)( 85,462)( 86,495)( 87,494)
( 88,510)( 89,509)( 90,508)( 91,507)( 92,506)( 93,505)( 94,504)( 95,503)
( 96,502)( 97,501)( 98,500)( 99,499)(100,498)(101,497)(102,496)(103,580)
(104,579)(105,595)(106,594)(107,593)(108,592)(109,591)(110,590)(111,589)
(112,588)(113,587)(114,586)(115,585)(116,584)(117,583)(118,582)(119,581)
(120,563)(121,562)(122,578)(123,577)(124,576)(125,575)(126,574)(127,573)
(128,572)(129,571)(130,570)(131,569)(132,568)(133,567)(134,566)(135,565)
(136,564)(137,597)(138,596)(139,612)(140,611)(141,610)(142,609)(143,608)
(144,607)(145,606)(146,605)(147,604)(148,603)(149,602)(150,601)(151,600)
(152,599)(153,598)(154,529)(155,528)(156,544)(157,543)(158,542)(159,541)
(160,540)(161,539)(162,538)(163,537)(164,536)(165,535)(166,534)(167,533)
(168,532)(169,531)(170,530)(171,512)(172,511)(173,527)(174,526)(175,525)
(176,524)(177,523)(178,522)(179,521)(180,520)(181,519)(182,518)(183,517)
(184,516)(185,515)(186,514)(187,513)(188,546)(189,545)(190,561)(191,560)
(192,559)(193,558)(194,557)(195,556)(196,555)(197,554)(198,553)(199,552)
(200,551)(201,550)(202,549)(203,548)(204,547)(205,733)(206,732)(207,748)
(208,747)(209,746)(210,745)(211,744)(212,743)(213,742)(214,741)(215,740)
(216,739)(217,738)(218,737)(219,736)(220,735)(221,734)(222,716)(223,715)
(224,731)(225,730)(226,729)(227,728)(228,727)(229,726)(230,725)(231,724)
(232,723)(233,722)(234,721)(235,720)(236,719)(237,718)(238,717)(239,750)
(240,749)(241,765)(242,764)(243,763)(244,762)(245,761)(246,760)(247,759)
(248,758)(249,757)(250,756)(251,755)(252,754)(253,753)(254,752)(255,751)
(256,784)(257,783)(258,799)(259,798)(260,797)(261,796)(262,795)(263,794)
(264,793)(265,792)(266,791)(267,790)(268,789)(269,788)(270,787)(271,786)
(272,785)(273,767)(274,766)(275,782)(276,781)(277,780)(278,779)(279,778)
(280,777)(281,776)(282,775)(283,774)(284,773)(285,772)(286,771)(287,770)
(288,769)(289,768)(290,801)(291,800)(292,816)(293,815)(294,814)(295,813)
(296,812)(297,811)(298,810)(299,809)(300,808)(301,807)(302,806)(303,805)
(304,804)(305,803)(306,802)(307,631)(308,630)(309,646)(310,645)(311,644)
(312,643)(313,642)(314,641)(315,640)(316,639)(317,638)(318,637)(319,636)
(320,635)(321,634)(322,633)(323,632)(324,614)(325,613)(326,629)(327,628)
(328,627)(329,626)(330,625)(331,624)(332,623)(333,622)(334,621)(335,620)
(336,619)(337,618)(338,617)(339,616)(340,615)(341,648)(342,647)(343,663)
(344,662)(345,661)(346,660)(347,659)(348,658)(349,657)(350,656)(351,655)
(352,654)(353,653)(354,652)(355,651)(356,650)(357,649)(358,682)(359,681)
(360,697)(361,696)(362,695)(363,694)(364,693)(365,692)(366,691)(367,690)
(368,689)(369,688)(370,687)(371,686)(372,685)(373,684)(374,683)(375,665)
(376,664)(377,680)(378,679)(379,678)(380,677)(381,676)(382,675)(383,674)
(384,673)(385,672)(386,671)(387,670)(388,669)(389,668)(390,667)(391,666)
(392,699)(393,698)(394,714)(395,713)(396,712)(397,711)(398,710)(399,709)
(400,708)(401,707)(402,706)(403,705)(404,704)(405,703)(406,702)(407,701)
(408,700);
poly := sub<Sym(816)|s0,s1>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope