#868700

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

1894 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,20)(19,24,23,21)(26,29)(28,31);(2,3)(6,7)(10,14)(15,16)(18,20)(19,23)(25,26,30,29)(27,31,32,28);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,15,14,16)(17,19)(18,21)(20,24)(22,23)(25,27,30,32)(26,31,29,28);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20,22,18)(19,24,23,21)(25,28)(26,27)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,14,15)(17,21,22,24)(18,23,20,19)(25,29,30,26)(27,28,32,31)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=ghi,d^2=h,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, 775896);