#621814

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

1522 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fghi,c^2=f,d^2=fg,e^2=h,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, 529010);