#624962

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

1568 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=fi,d^2=fh,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, 532158);