#6859328

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1567 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)(19,26)(20,29)(21,27)(22,24)(23,31)(25,28)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(18,24)(25,32)(26,29)(27,31);(10,14)(15,16)(17,19,22,20)(18,25,24,32)(21,28,23,30)(26,27,29,31);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,19)(18,26,24,29)(21,28,23,30)(25,27,32,31);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,24,31)(19,28,20,30)(25,29,32,26)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6766524);