#6867971

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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:

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

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,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, 6775167);