#725915

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=g,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 633111);