Polytope of Type {8,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*512b
if this polytope has a name.
Group : SmallGroup(512,32848)
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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      24 facets:
         16 of {4}*8
         8 of {8}*16
      16 vertex figures:
         16 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      16 facets:
         16 of {8}*16
      24 vertex figures:
         16 of {4}*8
         8 of {8}*16
   P/N, where N=<s0*s1*s0*s1> of order 4.
      20 facets:
         16 of {2}*4
         4 of {8}*16
      8 vertex figures:
         8 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 4.
      12 facets:
         8 of {4}*8
         4 of {8}*16
      8 vertex figures:
         8 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 4.
      8 facets:
         8 of {8}*16
      12 vertex figures:
         8 of {4}*8
         4 of {8}*16
   P/N, where N=<s1*s2*s1*s2> of order 4.
      8 facets:
         8 of {8}*16
      20 vertex figures:
         16 of {2}*4
         4 of {8}*16

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)( 17,149)( 18,150)( 19,152)( 20,151)( 21,145)( 22,146)( 23,148)( 24,147)( 25,157)( 26,158)( 27,160)( 28,159)( 29,153)( 30,154)( 31,156)( 32,155)( 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)( 49,189)( 50,190)( 51,192)( 52,191)( 53,185)( 54,186)( 55,188)( 56,187)( 57,181)( 58,182)( 59,184)( 60,183)( 61,177)( 62,178)( 63,180)( 64,179)( 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)( 81,213)( 82,214)( 83,216)( 84,215)( 85,209)( 86,210)( 87,212)( 88,211)( 89,221)( 90,222)( 91,224)( 92,223)( 93,217)( 94,218)( 95,220)( 96,219)( 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)(113,253)(114,254)(115,256)(116,255)(117,249)(118,250)(119,252)(120,251)(121,245)(122,246)(123,248)(124,247)(125,241)(126,242)(127,244)(128,243)(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)(273,405)(274,406)(275,408)(276,407)(277,401)(278,402)(279,404)(280,403)(281,413)(282,414)(283,416)(284,415)(285,409)(286,410)(287,412)(288,411)(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)(305,445)(306,446)(307,448)(308,447)(309,441)(310,442)(311,444)(312,443)(313,437)(314,438)(315,440)(316,439)(317,433)(318,434)(319,436)(320,435)(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)(337,469)(338,470)(339,472)(340,471)(341,465)(342,466)(343,468)(344,467)(345,477)(346,478)(347,480)(348,479)(349,473)(350,474)(351,476)(352,475)(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)(369,509)(370,510)(371,512)(372,511)(373,505)(374,506)(375,508)(376,507)(377,501)(378,502)(379,504)(380,503)(381,497)(382,498)(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)( 97,377)( 98,378)( 99,379)(100,380)(101,382)(102,381)(103,384)(104,383)(105,369)(106,370)(107,371)(108,372)(109,374)(110,373)(111,376)(112,375)(113,361)(114,362)(115,363)(116,364)(117,366)(118,365)(119,368)(120,367)(121,353)(122,354)(123,355)(124,356)(125,358)(126,357)(127,360)(128,359)(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)(197,497)(198,498)(199,499)(200,500)(201,509)(202,510)(203,511)(204,512)(205,505)(206,506)(207,507)(208,508)(209,485)(210,486)(211,487)(212,488)(213,481)(214,482)(215,483)(216,484)(217,493)(218,494)(219,495)(220,496)(221,489)(222,490)(223,491)(224,492)(225,469)(226,470)(227,471)(228,472)(229,465)(230,466)(231,467)(232,468)(233,477)(234,478)(235,479)(236,480)(237,473)(238,474)(239,475)(240,476)(241,453)(242,454)(243,455)(244,456)(245,449)(246,450)(247,451)(248,452)(249,461)(250,462)(251,463)(252,464)(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*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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)( 17,149)( 18,150)( 19,152)( 20,151)( 21,145)( 22,146)( 23,148)( 24,147)( 25,157)( 26,158)( 27,160)( 28,159)( 29,153)( 30,154)( 31,156)( 32,155)( 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)( 49,189)( 50,190)( 51,192)( 52,191)( 53,185)( 54,186)( 55,188)( 56,187)( 57,181)( 58,182)( 59,184)( 60,183)( 61,177)( 62,178)( 63,180)( 64,179)( 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)( 81,213)( 82,214)( 83,216)( 84,215)( 85,209)( 86,210)( 87,212)( 88,211)( 89,221)( 90,222)( 91,224)( 92,223)( 93,217)( 94,218)( 95,220)( 96,219)( 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)(113,253)(114,254)(115,256)(116,255)(117,249)(118,250)(119,252)(120,251)(121,245)(122,246)(123,248)(124,247)(125,241)(126,242)(127,244)(128,243)(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)(273,405)(274,406)(275,408)(276,407)(277,401)(278,402)(279,404)(280,403)(281,413)(282,414)(283,416)(284,415)(285,409)(286,410)(287,412)(288,411)(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)(305,445)(306,446)(307,448)(308,447)(309,441)(310,442)(311,444)(312,443)(313,437)(314,438)(315,440)(316,439)(317,433)(318,434)(319,436)(320,435)(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)(337,469)(338,470)(339,472)(340,471)(341,465)(342,466)(343,468)(344,467)(345,477)(346,478)(347,480)(348,479)(349,473)(350,474)(351,476)(352,475)(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)(369,509)(370,510)(371,512)(372,511)(373,505)(374,506)(375,508)(376,507)(377,501)(378,502)(379,504)(380,503)(381,497)(382,498)(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)( 97,377)( 98,378)( 99,379)(100,380)(101,382)(102,381)(103,384)(104,383)(105,369)(106,370)(107,371)(108,372)(109,374)(110,373)(111,376)(112,375)(113,361)(114,362)(115,363)(116,364)(117,366)(118,365)(119,368)(120,367)(121,353)(122,354)(123,355)(124,356)(125,358)(126,357)(127,360)(128,359)(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)(197,497)(198,498)(199,499)(200,500)(201,509)(202,510)(203,511)(204,512)(205,505)(206,506)(207,507)(208,508)(209,485)(210,486)(211,487)(212,488)(213,481)(214,482)(215,483)(216,484)(217,493)(218,494)(219,495)(220,496)(221,489)(222,490)(223,491)(224,492)(225,469)(226,470)(227,471)(228,472)(229,465)(230,466)(231,467)(232,468)(233,477)(234,478)(235,479)(236,480)(237,473)(238,474)(239,475)(240,476)(241,453)(242,454)(243,455)(244,456)(245,449)(246,450)(247,451)(248,452)(249,461)(250,462)(251,463)(252,464)(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*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope

Twisty Puzzle