Questions?
See the FAQ
or other info.

Polytope of Type {8,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*512e
if this polytope has a name.
Group : SmallGroup(512,58328)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,8}*256a
   4-fold quotients : {4,8}*128a, {4,4}*128, {4,8}*128b
   8-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
   16-fold quotients : {4,4}*32, {2,8}*32
   32-fold quotients : {2,4}*16, {4,2}*16
   64-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 74)( 10, 73)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 80)( 16, 79)
( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)( 24, 87)
( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,110)( 42,109)( 43,112)( 44,111)( 45,106)( 46,105)( 47,108)( 48,107)
( 49,118)( 50,117)( 51,120)( 52,119)( 53,114)( 54,113)( 55,116)( 56,115)
( 57,125)( 58,126)( 59,127)( 60,128)( 61,121)( 62,122)( 63,123)( 64,124)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,202)(138,201)(139,204)(140,203)(141,206)(142,205)(143,208)(144,207)
(145,210)(146,209)(147,212)(148,211)(149,214)(150,213)(151,216)(152,215)
(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,238)(170,237)(171,240)(172,239)(173,234)(174,233)(175,236)(176,235)
(177,246)(178,245)(179,248)(180,247)(181,242)(182,241)(183,244)(184,243)
(185,253)(186,254)(187,255)(188,256)(189,249)(190,250)(191,251)(192,252)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,330)(266,329)(267,332)(268,331)(269,334)(270,333)(271,336)(272,335)
(273,338)(274,337)(275,340)(276,339)(277,342)(278,341)(279,344)(280,343)
(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)(288,352)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,366)(298,365)(299,368)(300,367)(301,362)(302,361)(303,364)(304,363)
(305,374)(306,373)(307,376)(308,375)(309,370)(310,369)(311,372)(312,371)
(313,381)(314,382)(315,383)(316,384)(317,377)(318,378)(319,379)(320,380)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,458)(394,457)(395,460)(396,459)(397,462)(398,461)(399,464)(400,463)
(401,466)(402,465)(403,468)(404,467)(405,470)(406,469)(407,472)(408,471)
(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)(416,480)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,494)(426,493)(427,496)(428,495)(429,490)(430,489)(431,492)(432,491)
(433,502)(434,501)(435,504)(436,503)(437,498)(438,497)(439,500)(440,499)
(441,509)(442,510)(443,511)(444,512)(445,505)(446,506)(447,507)(448,508);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,263)(  6,264)(  7,261)(  8,262)
(  9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)( 16,270)
( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)( 40,302)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)( 48,294)
( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)( 56,317)
( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)( 64,309)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)( 72,342)
( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)( 80,350)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)( 96,334)
( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)(104,383)
(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)(112,375)
(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)(120,367)
(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)(128,359)
(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)(136,422)
(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)(144,430)
(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)(152,437)
(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)(160,445)
(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)(168,390)
(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)(176,398)
(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)(184,405)
(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)(192,413)
(193,501)(194,502)(195,503)(196,504)(197,499)(198,500)(199,497)(200,498)
(201,509)(202,510)(203,511)(204,512)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,487)(212,488)(213,483)(214,484)(215,481)(216,482)
(217,493)(218,494)(219,495)(220,496)(221,491)(222,492)(223,489)(224,490)
(225,471)(226,472)(227,469)(228,470)(229,465)(230,466)(231,467)(232,468)
(233,479)(234,480)(235,477)(236,478)(237,473)(238,474)(239,475)(240,476)
(241,455)(242,456)(243,453)(244,454)(245,449)(246,450)(247,451)(248,452)
(249,463)(250,464)(251,461)(252,462)(253,457)(254,458)(255,459)(256,460);;
s2 := (  1,129)(  2,130)(  3,132)(  4,131)(  5,134)(  6,133)(  7,135)(  8,136)
(  9,137)( 10,138)( 11,140)( 12,139)( 13,142)( 14,141)( 15,143)( 16,144)
( 17,151)( 18,152)( 19,150)( 20,149)( 21,148)( 22,147)( 23,145)( 24,146)
( 25,159)( 26,160)( 27,158)( 28,157)( 29,156)( 30,155)( 31,153)( 32,154)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,174)( 38,173)( 39,175)( 40,176)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,166)( 46,165)( 47,167)( 48,168)
( 49,191)( 50,192)( 51,190)( 52,189)( 53,188)( 54,187)( 55,185)( 56,186)
( 57,183)( 58,184)( 59,182)( 60,181)( 61,180)( 62,179)( 63,177)( 64,178)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,198)( 70,197)( 71,199)( 72,200)
( 73,201)( 74,202)( 75,204)( 76,203)( 77,206)( 78,205)( 79,207)( 80,208)
( 81,215)( 82,216)( 83,214)( 84,213)( 85,212)( 86,211)( 87,209)( 88,210)
( 89,223)( 90,224)( 91,222)( 92,221)( 93,220)( 94,219)( 95,217)( 96,218)
( 97,233)( 98,234)( 99,236)(100,235)(101,238)(102,237)(103,239)(104,240)
(105,225)(106,226)(107,228)(108,227)(109,230)(110,229)(111,231)(112,232)
(113,255)(114,256)(115,254)(116,253)(117,252)(118,251)(119,249)(120,250)
(121,247)(122,248)(123,246)(124,245)(125,244)(126,243)(127,241)(128,242)
(257,385)(258,386)(259,388)(260,387)(261,390)(262,389)(263,391)(264,392)
(265,393)(266,394)(267,396)(268,395)(269,398)(270,397)(271,399)(272,400)
(273,407)(274,408)(275,406)(276,405)(277,404)(278,403)(279,401)(280,402)
(281,415)(282,416)(283,414)(284,413)(285,412)(286,411)(287,409)(288,410)
(289,425)(290,426)(291,428)(292,427)(293,430)(294,429)(295,431)(296,432)
(297,417)(298,418)(299,420)(300,419)(301,422)(302,421)(303,423)(304,424)
(305,447)(306,448)(307,446)(308,445)(309,444)(310,443)(311,441)(312,442)
(313,439)(314,440)(315,438)(316,437)(317,436)(318,435)(319,433)(320,434)
(321,449)(322,450)(323,452)(324,451)(325,454)(326,453)(327,455)(328,456)
(329,457)(330,458)(331,460)(332,459)(333,462)(334,461)(335,463)(336,464)
(337,471)(338,472)(339,470)(340,469)(341,468)(342,467)(343,465)(344,466)
(345,479)(346,480)(347,478)(348,477)(349,476)(350,475)(351,473)(352,474)
(353,489)(354,490)(355,492)(356,491)(357,494)(358,493)(359,495)(360,496)
(361,481)(362,482)(363,484)(364,483)(365,486)(366,485)(367,487)(368,488)
(369,511)(370,512)(371,510)(372,509)(373,508)(374,507)(375,505)(376,506)
(377,503)(378,504)(379,502)(380,501)(381,500)(382,499)(383,497)(384,498);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(512)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)
(  8, 72)(  9, 74)( 10, 73)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 80)
( 16, 79)( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)
( 24, 87)( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,110)( 42,109)( 43,112)( 44,111)( 45,106)( 46,105)( 47,108)
( 48,107)( 49,118)( 50,117)( 51,120)( 52,119)( 53,114)( 54,113)( 55,116)
( 56,115)( 57,125)( 58,126)( 59,127)( 60,128)( 61,121)( 62,122)( 63,123)
( 64,124)(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)
(136,200)(137,202)(138,201)(139,204)(140,203)(141,206)(142,205)(143,208)
(144,207)(145,210)(146,209)(147,212)(148,211)(149,214)(150,213)(151,216)
(152,215)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)
(160,224)(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)
(168,228)(169,238)(170,237)(171,240)(172,239)(173,234)(174,233)(175,236)
(176,235)(177,246)(178,245)(179,248)(180,247)(181,242)(182,241)(183,244)
(184,243)(185,253)(186,254)(187,255)(188,256)(189,249)(190,250)(191,251)
(192,252)(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)
(264,328)(265,330)(266,329)(267,332)(268,331)(269,334)(270,333)(271,336)
(272,335)(273,338)(274,337)(275,340)(276,339)(277,342)(278,341)(279,344)
(280,343)(281,345)(282,346)(283,347)(284,348)(285,349)(286,350)(287,351)
(288,352)(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)
(296,356)(297,366)(298,365)(299,368)(300,367)(301,362)(302,361)(303,364)
(304,363)(305,374)(306,373)(307,376)(308,375)(309,370)(310,369)(311,372)
(312,371)(313,381)(314,382)(315,383)(316,384)(317,377)(318,378)(319,379)
(320,380)(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)
(392,456)(393,458)(394,457)(395,460)(396,459)(397,462)(398,461)(399,464)
(400,463)(401,466)(402,465)(403,468)(404,467)(405,470)(406,469)(407,472)
(408,471)(409,473)(410,474)(411,475)(412,476)(413,477)(414,478)(415,479)
(416,480)(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)
(424,484)(425,494)(426,493)(427,496)(428,495)(429,490)(430,489)(431,492)
(432,491)(433,502)(434,501)(435,504)(436,503)(437,498)(438,497)(439,500)
(440,499)(441,509)(442,510)(443,511)(444,512)(445,505)(446,506)(447,507)
(448,508);
s1 := Sym(512)!(  1,257)(  2,258)(  3,259)(  4,260)(  5,263)(  6,264)(  7,261)
(  8,262)(  9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)
( 16,270)( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)
( 24,277)( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)
( 40,302)( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)
( 48,294)( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)
( 56,317)( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)
( 64,309)( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)
( 72,342)( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)
( 80,350)( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)
( 88,326)( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)
( 96,334)( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)
(104,383)(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)
(112,375)(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)
(120,367)(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)
(128,359)(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)
(136,422)(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)
(144,430)(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)
(152,437)(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)
(160,445)(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)
(168,390)(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)
(176,398)(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)
(184,405)(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)
(192,413)(193,501)(194,502)(195,503)(196,504)(197,499)(198,500)(199,497)
(200,498)(201,509)(202,510)(203,511)(204,512)(205,507)(206,508)(207,505)
(208,506)(209,485)(210,486)(211,487)(212,488)(213,483)(214,484)(215,481)
(216,482)(217,493)(218,494)(219,495)(220,496)(221,491)(222,492)(223,489)
(224,490)(225,471)(226,472)(227,469)(228,470)(229,465)(230,466)(231,467)
(232,468)(233,479)(234,480)(235,477)(236,478)(237,473)(238,474)(239,475)
(240,476)(241,455)(242,456)(243,453)(244,454)(245,449)(246,450)(247,451)
(248,452)(249,463)(250,464)(251,461)(252,462)(253,457)(254,458)(255,459)
(256,460);
s2 := Sym(512)!(  1,129)(  2,130)(  3,132)(  4,131)(  5,134)(  6,133)(  7,135)
(  8,136)(  9,137)( 10,138)( 11,140)( 12,139)( 13,142)( 14,141)( 15,143)
( 16,144)( 17,151)( 18,152)( 19,150)( 20,149)( 21,148)( 22,147)( 23,145)
( 24,146)( 25,159)( 26,160)( 27,158)( 28,157)( 29,156)( 30,155)( 31,153)
( 32,154)( 33,169)( 34,170)( 35,172)( 36,171)( 37,174)( 38,173)( 39,175)
( 40,176)( 41,161)( 42,162)( 43,164)( 44,163)( 45,166)( 46,165)( 47,167)
( 48,168)( 49,191)( 50,192)( 51,190)( 52,189)( 53,188)( 54,187)( 55,185)
( 56,186)( 57,183)( 58,184)( 59,182)( 60,181)( 61,180)( 62,179)( 63,177)
( 64,178)( 65,193)( 66,194)( 67,196)( 68,195)( 69,198)( 70,197)( 71,199)
( 72,200)( 73,201)( 74,202)( 75,204)( 76,203)( 77,206)( 78,205)( 79,207)
( 80,208)( 81,215)( 82,216)( 83,214)( 84,213)( 85,212)( 86,211)( 87,209)
( 88,210)( 89,223)( 90,224)( 91,222)( 92,221)( 93,220)( 94,219)( 95,217)
( 96,218)( 97,233)( 98,234)( 99,236)(100,235)(101,238)(102,237)(103,239)
(104,240)(105,225)(106,226)(107,228)(108,227)(109,230)(110,229)(111,231)
(112,232)(113,255)(114,256)(115,254)(116,253)(117,252)(118,251)(119,249)
(120,250)(121,247)(122,248)(123,246)(124,245)(125,244)(126,243)(127,241)
(128,242)(257,385)(258,386)(259,388)(260,387)(261,390)(262,389)(263,391)
(264,392)(265,393)(266,394)(267,396)(268,395)(269,398)(270,397)(271,399)
(272,400)(273,407)(274,408)(275,406)(276,405)(277,404)(278,403)(279,401)
(280,402)(281,415)(282,416)(283,414)(284,413)(285,412)(286,411)(287,409)
(288,410)(289,425)(290,426)(291,428)(292,427)(293,430)(294,429)(295,431)
(296,432)(297,417)(298,418)(299,420)(300,419)(301,422)(302,421)(303,423)
(304,424)(305,447)(306,448)(307,446)(308,445)(309,444)(310,443)(311,441)
(312,442)(313,439)(314,440)(315,438)(316,437)(317,436)(318,435)(319,433)
(320,434)(321,449)(322,450)(323,452)(324,451)(325,454)(326,453)(327,455)
(328,456)(329,457)(330,458)(331,460)(332,459)(333,462)(334,461)(335,463)
(336,464)(337,471)(338,472)(339,470)(340,469)(341,468)(342,467)(343,465)
(344,466)(345,479)(346,480)(347,478)(348,477)(349,476)(350,475)(351,473)
(352,474)(353,489)(354,490)(355,492)(356,491)(357,494)(358,493)(359,495)
(360,496)(361,481)(362,482)(363,484)(364,483)(365,486)(366,485)(367,487)
(368,488)(369,511)(370,512)(371,510)(372,509)(373,508)(374,507)(375,505)
(376,506)(377,503)(378,504)(379,502)(380,501)(381,500)(382,499)(383,497)
(384,498);
poly := sub<Sym(512)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1 >; 
 
References : None.
to this polytope