#856505

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

1576 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=f,e^2=fg,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, 763701);