Polytope of Type {256}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {256}*512
Also Known As : 256-gon, {256}. if this polytope has another name.
Group : SmallGroup(512,2042)
Rank : 2
Schlafli Type : {256}
Number of vertices, edges, etc : 256, 256
Order of s₀s₁ : 256
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 : {128}*256
   4-fold quotients : {64}*128
   8-fold quotients : {32}*64
   16-fold quotients : {16}*32
   32-fold quotients : {8}*16
   64-fold quotients : {4}*8
   128-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,257)(  2,258)(  3,260)(  4,259)(  5,263)(  6,264)(  7,261)(  8,262)
(  9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)( 16,267)
( 17,281)( 18,282)( 19,284)( 20,283)( 21,287)( 22,288)( 23,285)( 24,286)
( 25,273)( 26,274)( 27,276)( 28,275)( 29,279)( 30,280)( 31,277)( 32,278)
( 33,305)( 34,306)( 35,308)( 36,307)( 37,311)( 38,312)( 39,309)( 40,310)
( 41,317)( 42,318)( 43,320)( 44,319)( 45,313)( 46,314)( 47,316)( 48,315)
( 49,289)( 50,290)( 51,292)( 52,291)( 53,295)( 54,296)( 55,293)( 56,294)
( 57,301)( 58,302)( 59,304)( 60,303)( 61,297)( 62,298)( 63,300)( 64,299)
( 65,353)( 66,354)( 67,356)( 68,355)( 69,359)( 70,360)( 71,357)( 72,358)
( 73,365)( 74,366)( 75,368)( 76,367)( 77,361)( 78,362)( 79,364)( 80,363)
( 81,377)( 82,378)( 83,380)( 84,379)( 85,383)( 86,384)( 87,381)( 88,382)
( 89,369)( 90,370)( 91,372)( 92,371)( 93,375)( 94,376)( 95,373)( 96,374)
( 97,321)( 98,322)( 99,324)(100,323)(101,327)(102,328)(103,325)(104,326)
(105,333)(106,334)(107,336)(108,335)(109,329)(110,330)(111,332)(112,331)
(113,345)(114,346)(115,348)(116,347)(117,351)(118,352)(119,349)(120,350)
(121,337)(122,338)(123,340)(124,339)(125,343)(126,344)(127,341)(128,342)
(129,449)(130,450)(131,452)(132,451)(133,455)(134,456)(135,453)(136,454)
(137,461)(138,462)(139,464)(140,463)(141,457)(142,458)(143,460)(144,459)
(145,473)(146,474)(147,476)(148,475)(149,479)(150,480)(151,477)(152,478)
(153,465)(154,466)(155,468)(156,467)(157,471)(158,472)(159,469)(160,470)
(161,497)(162,498)(163,500)(164,499)(165,503)(166,504)(167,501)(168,502)
(169,509)(170,510)(171,512)(172,511)(173,505)(174,506)(175,508)(176,507)
(177,481)(178,482)(179,484)(180,483)(181,487)(182,488)(183,485)(184,486)
(185,493)(186,494)(187,496)(188,495)(189,489)(190,490)(191,492)(192,491)
(193,385)(194,386)(195,388)(196,387)(197,391)(198,392)(199,389)(200,390)
(201,397)(202,398)(203,400)(204,399)(205,393)(206,394)(207,396)(208,395)
(209,409)(210,410)(211,412)(212,411)(213,415)(214,416)(215,413)(216,414)
(217,401)(218,402)(219,404)(220,403)(221,407)(222,408)(223,405)(224,406)
(225,433)(226,434)(227,436)(228,435)(229,439)(230,440)(231,437)(232,438)
(233,445)(234,446)(235,448)(236,447)(237,441)(238,442)(239,444)(240,443)
(241,417)(242,418)(243,420)(244,419)(245,423)(246,424)(247,421)(248,422)
(249,429)(250,430)(251,432)(252,431)(253,425)(254,426)(255,428)(256,427);;
s1 := (  1,129)(  2,130)(  3,132)(  4,131)(  5,135)(  6,136)(  7,133)(  8,134)
(  9,141)( 10,142)( 11,144)( 12,143)( 13,137)( 14,138)( 15,140)( 16,139)
( 17,153)( 18,154)( 19,156)( 20,155)( 21,159)( 22,160)( 23,157)( 24,158)
( 25,145)( 26,146)( 27,148)( 28,147)( 29,151)( 30,152)( 31,149)( 32,150)
( 33,177)( 34,178)( 35,180)( 36,179)( 37,183)( 38,184)( 39,181)( 40,182)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)( 48,187)
( 49,161)( 50,162)( 51,164)( 52,163)( 53,167)( 54,168)( 55,165)( 56,166)
( 57,173)( 58,174)( 59,176)( 60,175)( 61,169)( 62,170)( 63,172)( 64,171)
( 65,225)( 66,226)( 67,228)( 68,227)( 69,231)( 70,232)( 71,229)( 72,230)
( 73,237)( 74,238)( 75,240)( 76,239)( 77,233)( 78,234)( 79,236)( 80,235)
( 81,249)( 82,250)( 83,252)( 84,251)( 85,255)( 86,256)( 87,253)( 88,254)
( 89,241)( 90,242)( 91,244)( 92,243)( 93,247)( 94,248)( 95,245)( 96,246)
( 97,193)( 98,194)( 99,196)(100,195)(101,199)(102,200)(103,197)(104,198)
(105,205)(106,206)(107,208)(108,207)(109,201)(110,202)(111,204)(112,203)
(113,217)(114,218)(115,220)(116,219)(117,223)(118,224)(119,221)(120,222)
(121,209)(122,210)(123,212)(124,211)(125,215)(126,216)(127,213)(128,214)
(257,385)(258,386)(259,388)(260,387)(261,391)(262,392)(263,389)(264,390)
(265,397)(266,398)(267,400)(268,399)(269,393)(270,394)(271,396)(272,395)
(273,409)(274,410)(275,412)(276,411)(277,415)(278,416)(279,413)(280,414)
(281,401)(282,402)(283,404)(284,403)(285,407)(286,408)(287,405)(288,406)
(289,433)(290,434)(291,436)(292,435)(293,439)(294,440)(295,437)(296,438)
(297,445)(298,446)(299,448)(300,447)(301,441)(302,442)(303,444)(304,443)
(305,417)(306,418)(307,420)(308,419)(309,423)(310,424)(311,421)(312,422)
(313,429)(314,430)(315,432)(316,431)(317,425)(318,426)(319,428)(320,427)
(321,481)(322,482)(323,484)(324,483)(325,487)(326,488)(327,485)(328,486)
(329,493)(330,494)(331,496)(332,495)(333,489)(334,490)(335,492)(336,491)
(337,505)(338,506)(339,508)(340,507)(341,511)(342,512)(343,509)(344,510)
(345,497)(346,498)(347,500)(348,499)(349,503)(350,504)(351,501)(352,502)
(353,449)(354,450)(355,452)(356,451)(357,455)(358,456)(359,453)(360,454)
(361,461)(362,462)(363,464)(364,463)(365,457)(366,458)(367,460)(368,459)
(369,473)(370,474)(371,476)(372,475)(373,479)(374,480)(375,477)(376,478)
(377,465)(378,466)(379,468)(380,467)(381,471)(382,472)(383,469)(384,470);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(512)!(  1,257)(  2,258)(  3,260)(  4,259)(  5,263)(  6,264)(  7,261)
(  8,262)(  9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)
( 16,267)( 17,281)( 18,282)( 19,284)( 20,283)( 21,287)( 22,288)( 23,285)
( 24,286)( 25,273)( 26,274)( 27,276)( 28,275)( 29,279)( 30,280)( 31,277)
( 32,278)( 33,305)( 34,306)( 35,308)( 36,307)( 37,311)( 38,312)( 39,309)
( 40,310)( 41,317)( 42,318)( 43,320)( 44,319)( 45,313)( 46,314)( 47,316)
( 48,315)( 49,289)( 50,290)( 51,292)( 52,291)( 53,295)( 54,296)( 55,293)
( 56,294)( 57,301)( 58,302)( 59,304)( 60,303)( 61,297)( 62,298)( 63,300)
( 64,299)( 65,353)( 66,354)( 67,356)( 68,355)( 69,359)( 70,360)( 71,357)
( 72,358)( 73,365)( 74,366)( 75,368)( 76,367)( 77,361)( 78,362)( 79,364)
( 80,363)( 81,377)( 82,378)( 83,380)( 84,379)( 85,383)( 86,384)( 87,381)
( 88,382)( 89,369)( 90,370)( 91,372)( 92,371)( 93,375)( 94,376)( 95,373)
( 96,374)( 97,321)( 98,322)( 99,324)(100,323)(101,327)(102,328)(103,325)
(104,326)(105,333)(106,334)(107,336)(108,335)(109,329)(110,330)(111,332)
(112,331)(113,345)(114,346)(115,348)(116,347)(117,351)(118,352)(119,349)
(120,350)(121,337)(122,338)(123,340)(124,339)(125,343)(126,344)(127,341)
(128,342)(129,449)(130,450)(131,452)(132,451)(133,455)(134,456)(135,453)
(136,454)(137,461)(138,462)(139,464)(140,463)(141,457)(142,458)(143,460)
(144,459)(145,473)(146,474)(147,476)(148,475)(149,479)(150,480)(151,477)
(152,478)(153,465)(154,466)(155,468)(156,467)(157,471)(158,472)(159,469)
(160,470)(161,497)(162,498)(163,500)(164,499)(165,503)(166,504)(167,501)
(168,502)(169,509)(170,510)(171,512)(172,511)(173,505)(174,506)(175,508)
(176,507)(177,481)(178,482)(179,484)(180,483)(181,487)(182,488)(183,485)
(184,486)(185,493)(186,494)(187,496)(188,495)(189,489)(190,490)(191,492)
(192,491)(193,385)(194,386)(195,388)(196,387)(197,391)(198,392)(199,389)
(200,390)(201,397)(202,398)(203,400)(204,399)(205,393)(206,394)(207,396)
(208,395)(209,409)(210,410)(211,412)(212,411)(213,415)(214,416)(215,413)
(216,414)(217,401)(218,402)(219,404)(220,403)(221,407)(222,408)(223,405)
(224,406)(225,433)(226,434)(227,436)(228,435)(229,439)(230,440)(231,437)
(232,438)(233,445)(234,446)(235,448)(236,447)(237,441)(238,442)(239,444)
(240,443)(241,417)(242,418)(243,420)(244,419)(245,423)(246,424)(247,421)
(248,422)(249,429)(250,430)(251,432)(252,431)(253,425)(254,426)(255,428)
(256,427);
s1 := Sym(512)!(  1,129)(  2,130)(  3,132)(  4,131)(  5,135)(  6,136)(  7,133)
(  8,134)(  9,141)( 10,142)( 11,144)( 12,143)( 13,137)( 14,138)( 15,140)
( 16,139)( 17,153)( 18,154)( 19,156)( 20,155)( 21,159)( 22,160)( 23,157)
( 24,158)( 25,145)( 26,146)( 27,148)( 28,147)( 29,151)( 30,152)( 31,149)
( 32,150)( 33,177)( 34,178)( 35,180)( 36,179)( 37,183)( 38,184)( 39,181)
( 40,182)( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)
( 48,187)( 49,161)( 50,162)( 51,164)( 52,163)( 53,167)( 54,168)( 55,165)
( 56,166)( 57,173)( 58,174)( 59,176)( 60,175)( 61,169)( 62,170)( 63,172)
( 64,171)( 65,225)( 66,226)( 67,228)( 68,227)( 69,231)( 70,232)( 71,229)
( 72,230)( 73,237)( 74,238)( 75,240)( 76,239)( 77,233)( 78,234)( 79,236)
( 80,235)( 81,249)( 82,250)( 83,252)( 84,251)( 85,255)( 86,256)( 87,253)
( 88,254)( 89,241)( 90,242)( 91,244)( 92,243)( 93,247)( 94,248)( 95,245)
( 96,246)( 97,193)( 98,194)( 99,196)(100,195)(101,199)(102,200)(103,197)
(104,198)(105,205)(106,206)(107,208)(108,207)(109,201)(110,202)(111,204)
(112,203)(113,217)(114,218)(115,220)(116,219)(117,223)(118,224)(119,221)
(120,222)(121,209)(122,210)(123,212)(124,211)(125,215)(126,216)(127,213)
(128,214)(257,385)(258,386)(259,388)(260,387)(261,391)(262,392)(263,389)
(264,390)(265,397)(266,398)(267,400)(268,399)(269,393)(270,394)(271,396)
(272,395)(273,409)(274,410)(275,412)(276,411)(277,415)(278,416)(279,413)
(280,414)(281,401)(282,402)(283,404)(284,403)(285,407)(286,408)(287,405)
(288,406)(289,433)(290,434)(291,436)(292,435)(293,439)(294,440)(295,437)
(296,438)(297,445)(298,446)(299,448)(300,447)(301,441)(302,442)(303,444)
(304,443)(305,417)(306,418)(307,420)(308,419)(309,423)(310,424)(311,421)
(312,422)(313,429)(314,430)(315,432)(316,431)(317,425)(318,426)(319,428)
(320,427)(321,481)(322,482)(323,484)(324,483)(325,487)(326,488)(327,485)
(328,486)(329,493)(330,494)(331,496)(332,495)(333,489)(334,490)(335,492)
(336,491)(337,505)(338,506)(339,508)(340,507)(341,511)(342,512)(343,509)
(344,510)(345,497)(346,498)(347,500)(348,499)(349,503)(350,504)(351,501)
(352,502)(353,449)(354,450)(355,452)(356,451)(357,455)(358,456)(359,453)
(360,454)(361,461)(362,462)(363,464)(364,463)(365,457)(366,458)(367,460)
(368,459)(369,473)(370,474)(371,476)(372,475)(373,479)(374,480)(375,477)
(376,478)(377,465)(378,466)(379,468)(380,467)(381,471)(382,472)(383,469)
(384,470);
poly := sub<Sym(512)|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 >;

References : None.
to this polytope