#6757506

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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)(3,7)(4,6)(5,8)(17,18,24,31,22,20,23,27)(19,28,32,30,29,25,21,26);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,28)(21,23)(22,29)(24,32)(26,31)(27,30);(9,11,12,16)(10,13,14,15)(17,20,22,18)(19,28,29,25)(21,30,32,26)(23,27,24,31);(2,6)(5,7)(17,21,22,32)(18,26,20,30)(19,23,29,24)(25,31,28,27);(3,8)(5,7)(10,14)(13,15)(19,29)(21,32)(25,28)(26,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=hi,d^2=i,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, 6664702);