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D₅₁₂ #72210095

• p-group
• nilpotent
• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem
• ambivalent

Aliases

Subgroups

29 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 512 points:

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

Polycyclic presentation:

2^10|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,i^a=ij

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

Relationships

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);