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SD₅₁₂ #94847

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

Aliases

Subgroups

25 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 256 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);(2,128)(3,255)(4,126)(5,253)(6,124)(7,251)(8,122)(9,249)(10,120)(11,247)(12,118)(13,245)(14,116)(15,243)(16,114)(17,241)(18,112)(19,239)(20,110)(21,237)(22,108)(23,235)(24,106)(25,233)(26,104)(27,231)(28,102)(29,229)(30,100)(31,227)(32,98)(33,225)(34,96)(35,223)(36,94)(37,221)(38,92)(39,219)(40,90)(41,217)(42,88)(43,215)(44,86)(45,213)(46,84)(47,211)(48,82)(49,209)(50,80)(51,207)(52,78)(53,205)(54,76)(55,203)(56,74)(57,201)(58,72)(59,199)(60,70)(61,197)(62,68)(63,195)(64,66)(65,193)(67,191)(69,189)(71,187)(73,185)(75,183)(77,181)(79,179)(81,177)(83,175)(85,173)(87,171)(89,169)(91,167)(93,165)(95,163)(97,161)(99,159)(101,157)(103,155)(105,153)(107,151)(109,149)(111,147)(113,145)(115,143)(117,141)(119,139)(121,137)(123,135)(125,133)(127,131)(130,256)(132,254)(134,252)(136,250)(138,248)(140,246)(142,244)(144,242)(146,240)(148,238)(150,236)(152,234)(154,232)(156,230)(158,228)(160,226)(162,224)(164,222)(166,220)(168,218)(170,216)(172,214)(174,212)(176,210)(178,208)(180,206)(182,204)(184,202)(186,200)(188,198)(190,196)(192,194)

Polycyclic presentation:

2^9|a^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2043);