#619943

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 527139);