#6520136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427332);