#6660392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660385
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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2528 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,26)(21,25)(22,31)(23,29)(24,28)(30,32);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,23,25,27)(20,30,22,28)(24,31,32,26);(1,3)(2,6)(4,7)(5,8)(11,16)(13,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,32)(27,31);(2,5)(3,7)(10,14)(13,15)(18,25)(23,27)(24,32)(26,31);(2,5)(6,8)(17,20,21,22)(18,26,25,31)(19,28,29,30)(23,32,27,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567588);