#6690459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6597655);