#4335413

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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

1612 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=i,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4242609);