#6669388

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39335	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,19,27,36)(16,21,28,39)(17,30,32,22)(20,31,37,29)(24,38,25,34)(26,42,43,35)(33,41,44,40);(1,2)(3,4)(6,9)(7,11)(13,15)(14,19)(16,24)(17,26)(18,27)(20,33)(21,38)(22,35)(23,36)(25,28)(29,40)(30,42)(31,41)(32,43)(34,39)(37,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,24)(17,31)(18,28)(19,33)(21,30)(22,39)(23,37)(25,27)(26,41)(29,32)(34,35)(36,44)(38,42)(40,43);(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,28)(19,34)(20,30)(22,37)(23,39)(24,27)(26,41)(31,32)(33,35)(36,38)(40,43)(42,44);(2,4)(16,28)(20,37)(21,39)(24,25)(29,31)(33,44)(34,38)(40,41)

Polycyclic presentation:

2^9|a^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576584);