#6823402

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

1449 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fg,e^2=hi,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, 6730598);