#883874

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1646 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)(3,7)(17,18,20,22)(19,23,24,21)(25,26,29,32)(27,31,28,30);(1,2,6,5)(3,8,7,4)(9,10,13,12)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,27,29,28)(26,30,32,31);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15);(1,4,6,8)(2,7,5,3)(25,27,29,28)(26,30,32,31);(1,5,6,2)(3,8,7,4)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,21,22,23)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=gh,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 791070);