#845292

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1838 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)(6,8)(9,10)(11,13)(12,14)(15,16)(25,26,28,30)(27,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,21)(22,24);(2,5)(3,7)(11,15)(13,16)(25,27,28,32)(26,29,30,31);(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

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, 752488);