#633902

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1710 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)(6,8)(9,10,13,14)(11,15,16,12)(17,18,21,24)(19,20,23,22)(25,26,29,28)(27,30,32,31);(10,14)(11,16)(25,27,29,32)(26,30,28,31);(9,11,13,16)(10,15,14,12)(17,19,21,23)(18,22,24,20)(25,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24)(25,28,29,26)(27,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,20,21,22)(18,23,24,19)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fg,d^2=fh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 541098);