#624138

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1648 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)(3,5)(4,7)(6,8)(10,14)(11,16)(17,18,22,20)(19,21,24,23)(25,26,28,30)(27,29,31,32);(2,3)(6,7)(18,20)(19,24)(25,26,28,30)(27,29,31,32);(9,10,13,14)(11,15,16,12)(17,19,22,24)(18,23,20,21)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,20,22,18)(19,21,24,23);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(17,21)(18,19)(20,24)(22,23)(25,27,28,31)(26,29,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=fi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 531334);