#757791

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

1566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(9,10,14,13)(11,16,15,12)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(9,10,14,13)(11,12,15,16)(18,22)(23,24)(26,30)(31,33)(32,36)(39,40);(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,23,20,24)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,2,4,5)(3,6,7,8)(9,12,14,16)(10,15,13,11)(17,19,21,20)(18,23,22,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,15,12)(17,20,21,19)(18,24,22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fh,d^2=hi,e^2=fhi,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, 664987);