#855225

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

1730 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fghi,c^2=h,d^2=g,e^2=fh,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, 762421);