#153705

order	512
nilpotency class	7
isoclinism	isoclinic class of D₁₂₈
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂³
socle	C₂	C₂×D₆₄
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

75 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 256
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 256 points:

(1,2)(3,18)(4,11)(5,31)(6,38)(7,43)(8,46)(9,17)(10,20)(12,34)(13,33)(14,40)(15,29)(16,30)(19,73)(21,85)(22,89)(23,92)(24,94)(25,76)(26,95)(27,102)(28,107)(32,63)(35,127)(36,130)(37,115)(39,58)(41,142)(42,136)(44,74)(45,146)(47,75)(48,57)(49,77)(50,117)(51,116)(52,150)(53,84)(54,98)(55,97)(56,104)(59,165)(60,169)(61,172)(62,126)(64,177)(65,180)(66,123)(67,81)(68,184)(69,141)(70,82)(71,110)(72,83)(78,193)(79,195)(80,190)(86,200)(87,202)(88,199)(90,205)(91,204)(93,206)(96,160)(99,215)(100,113)(101,114)(103,157)(105,134)(106,135)(108,188)(109,145)(111,121)(112,124)(118,178)(119,181)(120,176)(122,133)(125,149)(128,174)(129,237)(131,175)(132,209)(137,166)(138,170)(139,173)(140,164)(143,163)(144,219)(147,203)(148,189)(151,211)(152,210)(153,238)(154,230)(155,228)(156,192)(158,243)(159,235)(161,233)(162,191)(167,249)(168,247)(171,251)(179,255)(182,227)(183,197)(185,196)(186,198)(187,229)(194,248)(201,250)(207,213)(208,214)(212,245)(216,242)(217,236)(218,226)(220,244)(221,240)(222,246)(223,232)(224,253)(225,234)(231,256)(239,254)(241,252);(1,3)(2,10)(4,19)(5,32)(6,39)(7,44)(8,47)(9,25)(11,49)(12,58)(13,63)(14,67)(15,70)(16,72)(17,53)(18,31)(20,34)(21,33)(22,40)(23,29)(24,30)(26,96)(27,103)(28,108)(35,128)(36,131)(37,120)(38,50)(41,143)(42,140)(43,51)(45,148)(46,52)(48,80)(54,157)(55,160)(56,162)(57,156)(59,166)(60,170)(61,173)(62,164)(64,178)(65,181)(66,176)(68,185)(69,183)(71,186)(73,95)(74,127)(75,130)(76,115)(77,98)(78,97)(79,104)(81,165)(82,169)(83,172)(84,126)(85,111)(86,177)(87,180)(88,123)(89,112)(90,184)(91,141)(92,113)(93,110)(94,114)(99,216)(100,118)(101,119)(102,151)(105,138)(106,139)(107,152)(109,147)(116,121)(117,124)(122,137)(125,150)(129,239)(132,212)(133,175)(134,174)(135,187)(136,154)(142,153)(144,220)(145,182)(146,155)(149,163)(158,200)(159,244)(161,245)(167,250)(168,248)(171,252)(179,256)(188,215)(189,237)(190,209)(191,243)(192,235)(193,207)(194,233)(195,208)(196,249)(197,247)(198,251)(199,223)(201,255)(202,222)(203,236)(204,225)(205,224)(206,226)(210,213)(211,214)(217,238)(218,231)(219,242)(221,241)(227,246)(228,232)(229,253)(230,234)(240,254);(1,4,9,30)(2,11,17,16)(3,19,25,24)(5,26,37,101)(6,27,42,106)(7,28,45,109)(8,29,48,110)(10,49,53,72)(12,54,62,61)(13,55,66,65)(14,56,69,68)(15,57,71,46)(18,73,76,94)(20,77,84,83)(21,78,88,87)(22,79,91,90)(23,80,93,47)(31,95,115,114)(32,96,120,119)(33,97,123,180)(34,98,126,172)(35,99,129,217)(36,100,132,218)(38,102,136,135)(39,103,140,139)(40,104,141,184)(41,105,144,221)(43,107,146,145)(44,108,148,147)(50,151,154,187)(51,152,155,182)(52,70,156,186)(58,157,164,173)(59,158,168,167)(60,159,171,149)(63,160,176,181)(64,161,179,133)(67,162,183,185)(74,188,189,203)(75,92,190,206)(81,191,197,196)(82,192,198,150)(85,193,199,202)(86,194,201,137)(89,195,204,205)(111,207,223,222)(112,208,225,224)(113,209,226,130)(116,210,228,227)(117,211,230,229)(118,212,231,131)(121,213,232,246)(122,177,233,255)(124,214,234,253)(125,169,235,251)(127,215,237,236)(128,216,239,238)(134,219,240,142)(138,220,241,143)(153,174,242,254)(163,170,244,252)(165,243,247,249)(166,200,248,250)(175,178,245,256)

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60901);