Polytope of Type {8,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*512a
if this polytope has a name.
Group : SmallGroup(512,32847)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Self-Petrie
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 : {8,8}*256a
4-fold quotients : {4,8}*128a, {8,4}*128a, {8,8}*128a, {8,8}*128b, {8,8}*128c, {8,8}*128d
8-fold quotients : {4,8}*64a, {8,4}*64a, {4,8}*64b, {8,4}*64b, {4,4}*64
16-fold quotients : {4,4}*32, {2,8}*32, {8,2}*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 := ( 1,129)( 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)
( 17,150)( 18,149)( 19,151)( 20,152)( 21,146)( 22,145)( 23,147)( 24,148)
( 25,158)( 26,157)( 27,159)( 28,160)( 29,154)( 30,153)( 31,155)( 32,156)
( 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)
( 49,190)( 50,189)( 51,191)( 52,192)( 53,186)( 54,185)( 55,187)( 56,188)
( 57,182)( 58,181)( 59,183)( 60,184)( 61,178)( 62,177)( 63,179)( 64,180)
( 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)
( 81,214)( 82,213)( 83,215)( 84,216)( 85,210)( 86,209)( 87,211)( 88,212)
( 89,222)( 90,221)( 91,223)( 92,224)( 93,218)( 94,217)( 95,219)( 96,220)
( 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)
(113,254)(114,253)(115,255)(116,256)(117,250)(118,249)(119,251)(120,252)
(121,246)(122,245)(123,247)(124,248)(125,242)(126,241)(127,243)(128,244)
(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)
(273,406)(274,405)(275,407)(276,408)(277,402)(278,401)(279,403)(280,404)
(281,414)(282,413)(283,415)(284,416)(285,410)(286,409)(287,411)(288,412)
(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)
(305,446)(306,445)(307,447)(308,448)(309,442)(310,441)(311,443)(312,444)
(313,438)(314,437)(315,439)(316,440)(317,434)(318,433)(319,435)(320,436)
(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)
(337,470)(338,469)(339,471)(340,472)(341,466)(342,465)(343,467)(344,468)
(345,478)(346,477)(347,479)(348,480)(349,474)(350,473)(351,475)(352,476)
(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)
(369,510)(370,509)(371,511)(372,512)(373,506)(374,505)(375,507)(376,508)
(377,502)(378,501)(379,503)(380,504)(381,498)(382,497)(383,499)(384,500);;
s1 := ( 1,257)( 2,258)( 3,259)( 4,260)( 5,261)( 6,262)( 7,263)( 8,264)
( 9,265)( 10,266)( 11,267)( 12,268)( 13,269)( 14,270)( 15,271)( 16,272)
( 17,275)( 18,276)( 19,273)( 20,274)( 21,279)( 22,280)( 23,277)( 24,278)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,287)( 30,288)( 31,285)( 32,286)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,301)( 38,302)( 39,303)( 40,304)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,293)( 46,294)( 47,295)( 48,296)
( 49,315)( 50,316)( 51,313)( 52,314)( 53,319)( 54,320)( 55,317)( 56,318)
( 57,307)( 58,308)( 59,305)( 60,306)( 61,311)( 62,312)( 63,309)( 64,310)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,341)( 70,342)( 71,343)( 72,344)
( 73,345)( 74,346)( 75,347)( 76,348)( 77,349)( 78,350)( 79,351)( 80,352)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,325)( 86,326)( 87,327)( 88,328)
( 89,329)( 90,330)( 91,331)( 92,332)( 93,333)( 94,334)( 95,335)( 96,336)
( 97,378)( 98,377)( 99,380)(100,379)(101,382)(102,381)(103,384)(104,383)
(105,370)(106,369)(107,372)(108,371)(109,374)(110,373)(111,376)(112,375)
(113,362)(114,361)(115,364)(116,363)(117,366)(118,365)(119,368)(120,367)
(121,354)(122,353)(123,356)(124,355)(125,358)(126,357)(127,360)(128,359)
(129,417)(130,418)(131,419)(132,420)(133,421)(134,422)(135,423)(136,424)
(137,425)(138,426)(139,427)(140,428)(141,429)(142,430)(143,431)(144,432)
(145,435)(146,436)(147,433)(148,434)(149,439)(150,440)(151,437)(152,438)
(153,443)(154,444)(155,441)(156,442)(157,447)(158,448)(159,445)(160,446)
(161,385)(162,386)(163,387)(164,388)(165,389)(166,390)(167,391)(168,392)
(169,393)(170,394)(171,395)(172,396)(173,397)(174,398)(175,399)(176,400)
(177,403)(178,404)(179,401)(180,402)(181,407)(182,408)(183,405)(184,406)
(185,411)(186,412)(187,409)(188,410)(189,415)(190,416)(191,413)(192,414)
(193,501)(194,502)(195,503)(196,504)(197,498)(198,497)(199,500)(200,499)
(201,509)(202,510)(203,511)(204,512)(205,506)(206,505)(207,508)(208,507)
(209,485)(210,486)(211,487)(212,488)(213,482)(214,481)(215,484)(216,483)
(217,493)(218,494)(219,495)(220,496)(221,490)(222,489)(223,492)(224,491)
(225,470)(226,469)(227,472)(228,471)(229,465)(230,466)(231,467)(232,468)
(233,478)(234,477)(235,480)(236,479)(237,473)(238,474)(239,475)(240,476)
(241,454)(242,453)(243,456)(244,455)(245,449)(246,450)(247,451)(248,452)
(249,462)(250,461)(251,464)(252,463)(253,457)(254,458)(255,459)(256,460);;
s2 := ( 1, 65)( 2, 66)( 3, 67)( 4, 68)( 5, 70)( 6, 69)( 7, 72)( 8, 71)
( 9, 74)( 10, 73)( 11, 76)( 12, 75)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 88)( 22, 87)( 23, 86)( 24, 85)
( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 95)( 30, 96)( 31, 93)( 32, 94)
( 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)
( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)( 56,114)
( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)( 64,121)
(129,193)(130,194)(131,195)(132,196)(133,198)(134,197)(135,200)(136,199)
(137,202)(138,201)(139,204)(140,203)(141,205)(142,206)(143,207)(144,208)
(145,211)(146,212)(147,209)(148,210)(149,216)(150,215)(151,214)(152,213)
(153,220)(154,219)(155,218)(156,217)(157,223)(158,224)(159,221)(160,222)
(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)
(177,247)(178,248)(179,245)(180,246)(181,243)(182,244)(183,241)(184,242)
(185,256)(186,255)(187,254)(188,253)(189,252)(190,251)(191,250)(192,249)
(257,321)(258,322)(259,323)(260,324)(261,326)(262,325)(263,328)(264,327)
(265,330)(266,329)(267,332)(268,331)(269,333)(270,334)(271,335)(272,336)
(273,339)(274,340)(275,337)(276,338)(277,344)(278,343)(279,342)(280,341)
(281,348)(282,347)(283,346)(284,345)(285,351)(286,352)(287,349)(288,350)
(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)
(305,375)(306,376)(307,373)(308,374)(309,371)(310,372)(311,369)(312,370)
(313,384)(314,383)(315,382)(316,381)(317,380)(318,379)(319,378)(320,377)
(385,449)(386,450)(387,451)(388,452)(389,454)(390,453)(391,456)(392,455)
(393,458)(394,457)(395,460)(396,459)(397,461)(398,462)(399,463)(400,464)
(401,467)(402,468)(403,465)(404,466)(405,472)(406,471)(407,470)(408,469)
(409,476)(410,475)(411,474)(412,473)(413,479)(414,480)(415,477)(416,478)
(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)
(433,503)(434,504)(435,501)(436,502)(437,499)(438,500)(439,497)(440,498)
(441,512)(442,511)(443,510)(444,509)(445,508)(446,507)(447,506)(448,505);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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)( 17,150)( 18,149)( 19,151)( 20,152)( 21,146)( 22,145)( 23,147)
( 24,148)( 25,158)( 26,157)( 27,159)( 28,160)( 29,154)( 30,153)( 31,155)
( 32,156)( 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)( 49,190)( 50,189)( 51,191)( 52,192)( 53,186)( 54,185)( 55,187)
( 56,188)( 57,182)( 58,181)( 59,183)( 60,184)( 61,178)( 62,177)( 63,179)
( 64,180)( 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)( 81,214)( 82,213)( 83,215)( 84,216)( 85,210)( 86,209)( 87,211)
( 88,212)( 89,222)( 90,221)( 91,223)( 92,224)( 93,218)( 94,217)( 95,219)
( 96,220)( 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)(113,254)(114,253)(115,255)(116,256)(117,250)(118,249)(119,251)
(120,252)(121,246)(122,245)(123,247)(124,248)(125,242)(126,241)(127,243)
(128,244)(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)(273,406)(274,405)(275,407)(276,408)(277,402)(278,401)(279,403)
(280,404)(281,414)(282,413)(283,415)(284,416)(285,410)(286,409)(287,411)
(288,412)(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)(305,446)(306,445)(307,447)(308,448)(309,442)(310,441)(311,443)
(312,444)(313,438)(314,437)(315,439)(316,440)(317,434)(318,433)(319,435)
(320,436)(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)(337,470)(338,469)(339,471)(340,472)(341,466)(342,465)(343,467)
(344,468)(345,478)(346,477)(347,479)(348,480)(349,474)(350,473)(351,475)
(352,476)(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)(369,510)(370,509)(371,511)(372,512)(373,506)(374,505)(375,507)
(376,508)(377,502)(378,501)(379,503)(380,504)(381,498)(382,497)(383,499)
(384,500);
s1 := Sym(512)!( 1,257)( 2,258)( 3,259)( 4,260)( 5,261)( 6,262)( 7,263)
( 8,264)( 9,265)( 10,266)( 11,267)( 12,268)( 13,269)( 14,270)( 15,271)
( 16,272)( 17,275)( 18,276)( 19,273)( 20,274)( 21,279)( 22,280)( 23,277)
( 24,278)( 25,283)( 26,284)( 27,281)( 28,282)( 29,287)( 30,288)( 31,285)
( 32,286)( 33,297)( 34,298)( 35,299)( 36,300)( 37,301)( 38,302)( 39,303)
( 40,304)( 41,289)( 42,290)( 43,291)( 44,292)( 45,293)( 46,294)( 47,295)
( 48,296)( 49,315)( 50,316)( 51,313)( 52,314)( 53,319)( 54,320)( 55,317)
( 56,318)( 57,307)( 58,308)( 59,305)( 60,306)( 61,311)( 62,312)( 63,309)
( 64,310)( 65,337)( 66,338)( 67,339)( 68,340)( 69,341)( 70,342)( 71,343)
( 72,344)( 73,345)( 74,346)( 75,347)( 76,348)( 77,349)( 78,350)( 79,351)
( 80,352)( 81,321)( 82,322)( 83,323)( 84,324)( 85,325)( 86,326)( 87,327)
( 88,328)( 89,329)( 90,330)( 91,331)( 92,332)( 93,333)( 94,334)( 95,335)
( 96,336)( 97,378)( 98,377)( 99,380)(100,379)(101,382)(102,381)(103,384)
(104,383)(105,370)(106,369)(107,372)(108,371)(109,374)(110,373)(111,376)
(112,375)(113,362)(114,361)(115,364)(116,363)(117,366)(118,365)(119,368)
(120,367)(121,354)(122,353)(123,356)(124,355)(125,358)(126,357)(127,360)
(128,359)(129,417)(130,418)(131,419)(132,420)(133,421)(134,422)(135,423)
(136,424)(137,425)(138,426)(139,427)(140,428)(141,429)(142,430)(143,431)
(144,432)(145,435)(146,436)(147,433)(148,434)(149,439)(150,440)(151,437)
(152,438)(153,443)(154,444)(155,441)(156,442)(157,447)(158,448)(159,445)
(160,446)(161,385)(162,386)(163,387)(164,388)(165,389)(166,390)(167,391)
(168,392)(169,393)(170,394)(171,395)(172,396)(173,397)(174,398)(175,399)
(176,400)(177,403)(178,404)(179,401)(180,402)(181,407)(182,408)(183,405)
(184,406)(185,411)(186,412)(187,409)(188,410)(189,415)(190,416)(191,413)
(192,414)(193,501)(194,502)(195,503)(196,504)(197,498)(198,497)(199,500)
(200,499)(201,509)(202,510)(203,511)(204,512)(205,506)(206,505)(207,508)
(208,507)(209,485)(210,486)(211,487)(212,488)(213,482)(214,481)(215,484)
(216,483)(217,493)(218,494)(219,495)(220,496)(221,490)(222,489)(223,492)
(224,491)(225,470)(226,469)(227,472)(228,471)(229,465)(230,466)(231,467)
(232,468)(233,478)(234,477)(235,480)(236,479)(237,473)(238,474)(239,475)
(240,476)(241,454)(242,453)(243,456)(244,455)(245,449)(246,450)(247,451)
(248,452)(249,462)(250,461)(251,464)(252,463)(253,457)(254,458)(255,459)
(256,460);
s2 := Sym(512)!( 1, 65)( 2, 66)( 3, 67)( 4, 68)( 5, 70)( 6, 69)( 7, 72)
( 8, 71)( 9, 74)( 10, 73)( 11, 76)( 12, 75)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 88)( 22, 87)( 23, 86)
( 24, 85)( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 95)( 30, 96)( 31, 93)
( 32, 94)( 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)( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)
( 56,114)( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)
( 64,121)(129,193)(130,194)(131,195)(132,196)(133,198)(134,197)(135,200)
(136,199)(137,202)(138,201)(139,204)(140,203)(141,205)(142,206)(143,207)
(144,208)(145,211)(146,212)(147,209)(148,210)(149,216)(150,215)(151,214)
(152,213)(153,220)(154,219)(155,218)(156,217)(157,223)(158,224)(159,221)
(160,222)(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)(177,247)(178,248)(179,245)(180,246)(181,243)(182,244)(183,241)
(184,242)(185,256)(186,255)(187,254)(188,253)(189,252)(190,251)(191,250)
(192,249)(257,321)(258,322)(259,323)(260,324)(261,326)(262,325)(263,328)
(264,327)(265,330)(266,329)(267,332)(268,331)(269,333)(270,334)(271,335)
(272,336)(273,339)(274,340)(275,337)(276,338)(277,344)(278,343)(279,342)
(280,341)(281,348)(282,347)(283,346)(284,345)(285,351)(286,352)(287,349)
(288,350)(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)(305,375)(306,376)(307,373)(308,374)(309,371)(310,372)(311,369)
(312,370)(313,384)(314,383)(315,382)(316,381)(317,380)(318,379)(319,378)
(320,377)(385,449)(386,450)(387,451)(388,452)(389,454)(390,453)(391,456)
(392,455)(393,458)(394,457)(395,460)(396,459)(397,461)(398,462)(399,463)
(400,464)(401,467)(402,468)(403,465)(404,466)(405,472)(406,471)(407,470)
(408,469)(409,476)(410,475)(411,474)(412,473)(413,479)(414,480)(415,477)
(416,478)(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)(433,503)(434,504)(435,501)(436,502)(437,499)(438,500)(439,497)
(440,498)(441,512)(442,511)(443,510)(444,509)(445,508)(446,507)(447,506)
(448,505);
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*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope