#6832576

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,24,30)(19,25,20,26)(21,27,23,29)(22,28,31,32);(9,11)(10,13)(12,16)(14,15)(17,19,24,20)(18,25,30,26)(21,22,23,31)(27,28,29,32);(2,5)(6,8)(10,14)(11,16)(17,20,23,31,24,19,21,22)(18,26,29,32,30,25,27,28);(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,22,20,31)(25,28,26,32);(1,3,4,7)(2,6,5,8)(17,22,23,20,24,31,21,19)(18,28,29,26,30,32,27,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fh,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739772);