#732875

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gi,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 640071);