#751631

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

1844 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fg,d^2=h,e^2=hi,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, 658827);