#509594

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

692 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)(11,16,21,20)(12,24,27,40)(13,15,28,30)(17,33,34,25)(18,26,31,23)(22,37,36,29)(32,38,35,39);(5,6)(7,8)(10,15)(11,21)(12,25)(13,23)(16,31)(18,20)(19,30)(22,38)(24,29)(26,28)(27,33)(32,35)(36,39)(37,40);(2,4)(6,8)(9,11,14,21)(10,16,19,20)(12,22,27,36)(13,23,28,26)(15,18,30,31)(17,32,34,35)(24,37,40,29)(25,39,33,38);(2,4)(9,12,26,38,11,22,13,25,14,27,23,39,21,36,28,33)(10,17,30,37,20,35,18,24,19,34,15,29,16,32,31,40)

Polycyclic presentation:

2^9|a^2=i,c^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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416790);