#952367

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1590 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 859563);