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Q₅₁₂ #94848

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

Aliases

Subgroups

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

131 conjugacy classes — view the full character table.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2044);