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# Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*512c
if this polytope has a name.
Group : SmallGroup(512,32850)
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
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, {4,8}*256a, {8,4}*256a, {8,8}*256b, {8,8}*256c, {8,8}*256d
4-fold quotients : {4,8}*128a, {8,4}*128a, {8,8}*128a, {8,8}*128b, {8,8}*128c, {8,8}*128d, {4,4}*128, {4,8}*128b, {8,4}*128b
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)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)(  8,136)
(  9,137)( 10,138)( 11,139)( 12,140)( 13,141)( 14,142)( 15,143)( 16,144)
( 17,150)( 18,149)( 19,152)( 20,151)( 21,146)( 22,145)( 23,148)( 24,147)
( 25,158)( 26,157)( 27,160)( 28,159)( 29,154)( 30,153)( 31,156)( 32,155)
( 33,169)( 34,170)( 35,171)( 36,172)( 37,173)( 38,174)( 39,175)( 40,176)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,190)( 50,189)( 51,192)( 52,191)( 53,186)( 54,185)( 55,188)( 56,187)
( 57,182)( 58,181)( 59,184)( 60,183)( 61,178)( 62,177)( 63,180)( 64,179)
( 65,193)( 66,194)( 67,195)( 68,196)( 69,197)( 70,198)( 71,199)( 72,200)
( 73,201)( 74,202)( 75,203)( 76,204)( 77,205)( 78,206)( 79,207)( 80,208)
( 81,214)( 82,213)( 83,216)( 84,215)( 85,210)( 86,209)( 87,212)( 88,211)
( 89,222)( 90,221)( 91,224)( 92,223)( 93,218)( 94,217)( 95,220)( 96,219)
( 97,233)( 98,234)( 99,235)(100,236)(101,237)(102,238)(103,239)(104,240)
(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)
(113,254)(114,253)(115,256)(116,255)(117,250)(118,249)(119,252)(120,251)
(121,246)(122,245)(123,248)(124,247)(125,242)(126,241)(127,244)(128,243)
(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)
(265,393)(266,394)(267,395)(268,396)(269,397)(270,398)(271,399)(272,400)
(273,406)(274,405)(275,408)(276,407)(277,402)(278,401)(279,404)(280,403)
(281,414)(282,413)(283,416)(284,415)(285,410)(286,409)(287,412)(288,411)
(289,425)(290,426)(291,427)(292,428)(293,429)(294,430)(295,431)(296,432)
(297,417)(298,418)(299,419)(300,420)(301,421)(302,422)(303,423)(304,424)
(305,446)(306,445)(307,448)(308,447)(309,442)(310,441)(311,444)(312,443)
(313,438)(314,437)(315,440)(316,439)(317,434)(318,433)(319,436)(320,435)
(321,449)(322,450)(323,451)(324,452)(325,453)(326,454)(327,455)(328,456)
(329,457)(330,458)(331,459)(332,460)(333,461)(334,462)(335,463)(336,464)
(337,470)(338,469)(339,472)(340,471)(341,466)(342,465)(343,468)(344,467)
(345,478)(346,477)(347,480)(348,479)(349,474)(350,473)(351,476)(352,475)
(353,489)(354,490)(355,491)(356,492)(357,493)(358,494)(359,495)(360,496)
(361,481)(362,482)(363,483)(364,484)(365,485)(366,486)(367,487)(368,488)
(369,510)(370,509)(371,512)(372,511)(373,506)(374,505)(375,508)(376,507)
(377,502)(378,501)(379,504)(380,503)(381,498)(382,497)(383,500)(384,499);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)(  8,263)
(  9,265)( 10,266)( 11,267)( 12,268)( 13,270)( 14,269)( 15,272)( 16,271)
( 17,275)( 18,276)( 19,273)( 20,274)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,302)( 38,301)( 39,304)( 40,303)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,294)( 46,293)( 47,296)( 48,295)
( 49,315)( 50,316)( 51,313)( 52,314)( 53,320)( 54,319)( 55,318)( 56,317)
( 57,307)( 58,308)( 59,305)( 60,306)( 61,312)( 62,311)( 63,310)( 64,309)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,342)( 70,341)( 71,344)( 72,343)
( 73,345)( 74,346)( 75,347)( 76,348)( 77,350)( 78,349)( 79,352)( 80,351)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,326)( 86,325)( 87,328)( 88,327)
( 89,329)( 90,330)( 91,331)( 92,332)( 93,334)( 94,333)( 95,336)( 96,335)
( 97,378)( 98,377)( 99,380)(100,379)(101,381)(102,382)(103,383)(104,384)
(105,370)(106,369)(107,372)(108,371)(109,373)(110,374)(111,375)(112,376)
(113,362)(114,361)(115,364)(116,363)(117,365)(118,366)(119,367)(120,368)
(121,354)(122,353)(123,356)(124,355)(125,357)(126,358)(127,359)(128,360)
(129,417)(130,418)(131,419)(132,420)(133,422)(134,421)(135,424)(136,423)
(137,425)(138,426)(139,427)(140,428)(141,430)(142,429)(143,432)(144,431)
(145,435)(146,436)(147,433)(148,434)(149,440)(150,439)(151,438)(152,437)
(153,443)(154,444)(155,441)(156,442)(157,448)(158,447)(159,446)(160,445)
(161,385)(162,386)(163,387)(164,388)(165,390)(166,389)(167,392)(168,391)
(169,393)(170,394)(171,395)(172,396)(173,398)(174,397)(175,400)(176,399)
(177,403)(178,404)(179,401)(180,402)(181,408)(182,407)(183,406)(184,405)
(185,411)(186,412)(187,409)(188,410)(189,416)(190,415)(191,414)(192,413)
(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, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 87)( 22, 88)( 23, 85)( 24, 86)
( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 95)( 30, 96)( 31, 93)( 32, 94)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,109)( 42,110)( 43,111)( 44,112)( 45,105)( 46,106)( 47,107)( 48,108)
( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)( 56,114)
( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)( 64,122)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)
(145,211)(146,212)(147,209)(148,210)(149,215)(150,216)(151,213)(152,214)
(153,219)(154,220)(155,217)(156,218)(157,223)(158,224)(159,221)(160,222)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)
(177,247)(178,248)(179,245)(180,246)(181,243)(182,244)(183,241)(184,242)
(185,255)(186,256)(187,253)(188,254)(189,251)(190,252)(191,249)(192,250)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)(272,336)
(273,339)(274,340)(275,337)(276,338)(277,343)(278,344)(279,341)(280,342)
(281,347)(282,348)(283,345)(284,346)(285,351)(286,352)(287,349)(288,350)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,365)(298,366)(299,367)(300,368)(301,361)(302,362)(303,363)(304,364)
(305,375)(306,376)(307,373)(308,374)(309,371)(310,372)(311,369)(312,370)
(313,383)(314,384)(315,381)(316,382)(317,379)(318,380)(319,377)(320,378)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)(400,464)
(401,467)(402,468)(403,465)(404,466)(405,471)(406,472)(407,469)(408,470)
(409,475)(410,476)(411,473)(412,474)(413,479)(414,480)(415,477)(416,478)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,493)(426,494)(427,495)(428,496)(429,489)(430,490)(431,491)(432,492)
(433,503)(434,504)(435,501)(436,502)(437,499)(438,500)(439,497)(440,498)
(441,511)(442,512)(443,509)(444,510)(445,507)(446,508)(447,505)(448,506);;
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,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*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)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)
(  8,136)(  9,137)( 10,138)( 11,139)( 12,140)( 13,141)( 14,142)( 15,143)
( 16,144)( 17,150)( 18,149)( 19,152)( 20,151)( 21,146)( 22,145)( 23,148)
( 24,147)( 25,158)( 26,157)( 27,160)( 28,159)( 29,154)( 30,153)( 31,156)
( 32,155)( 33,169)( 34,170)( 35,171)( 36,172)( 37,173)( 38,174)( 39,175)
( 40,176)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,190)( 50,189)( 51,192)( 52,191)( 53,186)( 54,185)( 55,188)
( 56,187)( 57,182)( 58,181)( 59,184)( 60,183)( 61,178)( 62,177)( 63,180)
( 64,179)( 65,193)( 66,194)( 67,195)( 68,196)( 69,197)( 70,198)( 71,199)
( 72,200)( 73,201)( 74,202)( 75,203)( 76,204)( 77,205)( 78,206)( 79,207)
( 80,208)( 81,214)( 82,213)( 83,216)( 84,215)( 85,210)( 86,209)( 87,212)
( 88,211)( 89,222)( 90,221)( 91,224)( 92,223)( 93,218)( 94,217)( 95,220)
( 96,219)( 97,233)( 98,234)( 99,235)(100,236)(101,237)(102,238)(103,239)
(104,240)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)
(112,232)(113,254)(114,253)(115,256)(116,255)(117,250)(118,249)(119,252)
(120,251)(121,246)(122,245)(123,248)(124,247)(125,242)(126,241)(127,244)
(128,243)(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)
(264,392)(265,393)(266,394)(267,395)(268,396)(269,397)(270,398)(271,399)
(272,400)(273,406)(274,405)(275,408)(276,407)(277,402)(278,401)(279,404)
(280,403)(281,414)(282,413)(283,416)(284,415)(285,410)(286,409)(287,412)
(288,411)(289,425)(290,426)(291,427)(292,428)(293,429)(294,430)(295,431)
(296,432)(297,417)(298,418)(299,419)(300,420)(301,421)(302,422)(303,423)
(304,424)(305,446)(306,445)(307,448)(308,447)(309,442)(310,441)(311,444)
(312,443)(313,438)(314,437)(315,440)(316,439)(317,434)(318,433)(319,436)
(320,435)(321,449)(322,450)(323,451)(324,452)(325,453)(326,454)(327,455)
(328,456)(329,457)(330,458)(331,459)(332,460)(333,461)(334,462)(335,463)
(336,464)(337,470)(338,469)(339,472)(340,471)(341,466)(342,465)(343,468)
(344,467)(345,478)(346,477)(347,480)(348,479)(349,474)(350,473)(351,476)
(352,475)(353,489)(354,490)(355,491)(356,492)(357,493)(358,494)(359,495)
(360,496)(361,481)(362,482)(363,483)(364,484)(365,485)(366,486)(367,487)
(368,488)(369,510)(370,509)(371,512)(372,511)(373,506)(374,505)(375,508)
(376,507)(377,502)(378,501)(379,504)(380,503)(381,498)(382,497)(383,500)
(384,499);
s1 := Sym(512)!(  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)
(  8,263)(  9,265)( 10,266)( 11,267)( 12,268)( 13,270)( 14,269)( 15,272)
( 16,271)( 17,275)( 18,276)( 19,273)( 20,274)( 21,280)( 22,279)( 23,278)
( 24,277)( 25,283)( 26,284)( 27,281)( 28,282)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,297)( 34,298)( 35,299)( 36,300)( 37,302)( 38,301)( 39,304)
( 40,303)( 41,289)( 42,290)( 43,291)( 44,292)( 45,294)( 46,293)( 47,296)
( 48,295)( 49,315)( 50,316)( 51,313)( 52,314)( 53,320)( 54,319)( 55,318)
( 56,317)( 57,307)( 58,308)( 59,305)( 60,306)( 61,312)( 62,311)( 63,310)
( 64,309)( 65,337)( 66,338)( 67,339)( 68,340)( 69,342)( 70,341)( 71,344)
( 72,343)( 73,345)( 74,346)( 75,347)( 76,348)( 77,350)( 78,349)( 79,352)
( 80,351)( 81,321)( 82,322)( 83,323)( 84,324)( 85,326)( 86,325)( 87,328)
( 88,327)( 89,329)( 90,330)( 91,331)( 92,332)( 93,334)( 94,333)( 95,336)
( 96,335)( 97,378)( 98,377)( 99,380)(100,379)(101,381)(102,382)(103,383)
(104,384)(105,370)(106,369)(107,372)(108,371)(109,373)(110,374)(111,375)
(112,376)(113,362)(114,361)(115,364)(116,363)(117,365)(118,366)(119,367)
(120,368)(121,354)(122,353)(123,356)(124,355)(125,357)(126,358)(127,359)
(128,360)(129,417)(130,418)(131,419)(132,420)(133,422)(134,421)(135,424)
(136,423)(137,425)(138,426)(139,427)(140,428)(141,430)(142,429)(143,432)
(144,431)(145,435)(146,436)(147,433)(148,434)(149,440)(150,439)(151,438)
(152,437)(153,443)(154,444)(155,441)(156,442)(157,448)(158,447)(159,446)
(160,445)(161,385)(162,386)(163,387)(164,388)(165,390)(166,389)(167,392)
(168,391)(169,393)(170,394)(171,395)(172,396)(173,398)(174,397)(175,400)
(176,399)(177,403)(178,404)(179,401)(180,402)(181,408)(182,407)(183,406)
(184,405)(185,411)(186,412)(187,409)(188,410)(189,416)(190,415)(191,414)
(192,413)(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, 69)(  6, 70)(  7, 71)
(  8, 72)(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 83)( 18, 84)( 19, 81)( 20, 82)( 21, 87)( 22, 88)( 23, 85)
( 24, 86)( 25, 91)( 26, 92)( 27, 89)( 28, 90)( 29, 95)( 30, 96)( 31, 93)
( 32, 94)( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,109)( 42,110)( 43,111)( 44,112)( 45,105)( 46,106)( 47,107)
( 48,108)( 49,119)( 50,120)( 51,117)( 52,118)( 53,115)( 54,116)( 55,113)
( 56,114)( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)
( 64,122)(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)
(136,200)(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)
(144,208)(145,211)(146,212)(147,209)(148,210)(149,215)(150,216)(151,213)
(152,214)(153,219)(154,220)(155,217)(156,218)(157,223)(158,224)(159,221)
(160,222)(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)
(168,228)(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)
(176,236)(177,247)(178,248)(179,245)(180,246)(181,243)(182,244)(183,241)
(184,242)(185,255)(186,256)(187,253)(188,254)(189,251)(190,252)(191,249)
(192,250)(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)
(264,328)(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)
(272,336)(273,339)(274,340)(275,337)(276,338)(277,343)(278,344)(279,341)
(280,342)(281,347)(282,348)(283,345)(284,346)(285,351)(286,352)(287,349)
(288,350)(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)
(296,356)(297,365)(298,366)(299,367)(300,368)(301,361)(302,362)(303,363)
(304,364)(305,375)(306,376)(307,373)(308,374)(309,371)(310,372)(311,369)
(312,370)(313,383)(314,384)(315,381)(316,382)(317,379)(318,380)(319,377)
(320,378)(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)
(392,456)(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)
(400,464)(401,467)(402,468)(403,465)(404,466)(405,471)(406,472)(407,469)
(408,470)(409,475)(410,476)(411,473)(412,474)(413,479)(414,480)(415,477)
(416,478)(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)
(424,484)(425,493)(426,494)(427,495)(428,496)(429,489)(430,490)(431,491)
(432,492)(433,503)(434,504)(435,501)(436,502)(437,499)(438,500)(439,497)
(440,498)(441,511)(442,512)(443,509)(444,510)(445,507)(446,508)(447,505)
(448,506);
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,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope