#6867394

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#31036	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774590);