#509602

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416798);