#6823399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fgi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6730595);