#6761245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=h,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6668441);