#616502

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

1942 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 523698);