#760813

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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

1758 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=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, 668009);