#758739

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1630 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=g,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 665935);