#501603

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 408799);