#6632663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632008
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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,26,27,32)(20,25,22,29)(24,30,31,28);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,27)(25,29)(28,30);(11,15)(13,16)(17,19)(18,24)(20,28)(21,27)(22,30)(23,31)(25,32)(26,29);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,23)(24,31)(25,29)(26,32);(2,5)(6,7)(10,14)(13,16)(17,20,21,22)(18,25,23,29)(19,28,27,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539859);