Polytope of Type {688}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {688}*1376
Also Known As : 688-gon, {688}. if this polytope has another name.
Group : SmallGroup(1376,88)
Rank : 2
Schlafli Type : {688}
Number of vertices, edges, etc : 688, 688
Order of s₀s₁ : 688
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 : {344}*688
   4-fold quotients : {172}*344
   8-fold quotients : {86}*172
   16-fold quotients : {43}*86
   43-fold quotients : {16}*32
   86-fold quotients : {8}*16
   172-fold quotients : {4}*8
   344-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope