#6796182

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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)(3,5)(4,7)(6,8)(17,18,24,32,23,26,22,25)(19,20,27,29,21,31,28,30);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,19,23,21)(18,20,26,31)(22,28,24,27)(25,30,32,29);(2,3)(6,7)(10,13)(14,15)(17,20,23,31)(18,21,26,19)(22,29,24,30)(25,28,32,27);(1,4,5,8)(2,6,3,7)(17,21,23,19)(18,20,26,31)(22,27,24,28)(25,30,32,29);(9,11,12,16)(10,14,13,15)(17,22,23,24)(18,25,26,32)(19,27,21,28)(20,29,31,30)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,d^2=fi,e^2=fh,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, 6703378);