#659268

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1974 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)(3,6)(4,5)(7,8)(9,10,14,13)(11,16,15,12)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,32)(29,30);(1,3,7,5)(2,4,8,6)(10,13)(11,15)(18,20)(22,23)(26,30)(28,31);(1,4,7,6)(2,5,8,3)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27,29,32)(26,31,30,28);(1,5,7,3)(2,4,8,6)(9,12,14,16)(10,15,13,11);(1,6,7,4)(2,3,8,5)(9,13,14,10)(11,16,15,12)(17,20,21,18)(19,23,24,22)(25,28,29,31)(26,32,30,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 566464);