#868702

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

1726 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=fghi,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, 775898);