#6867939

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 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:

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

Polycyclic presentation:

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

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, 6775135);