#880589

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1814 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 787785);