#950614

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1744 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=fg,d^2=fgi,e^2=fh,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, 857810);