#509142

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

855 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)(9,10,13,19,11,17,23,32,14,20,29,33,15,21,27,16)(12,24,25,34,30,37,39,18,26,31,40,35,22,38,28,36);(5,6)(7,8)(10,16)(11,15)(12,25)(13,27)(17,33)(18,35)(19,21)(20,32)(22,39)(23,29)(26,40)(28,30)(34,36)(37,38);(2,4)(6,8)(9,11,14,15)(10,17,20,21)(12,22,26,30)(13,23,29,27)(16,19,32,33)(18,34,36,35)(24,38,31,37)(25,28,40,39);(2,4)(9,12,14,26)(10,18,20,36)(11,22,15,30)(13,28,29,39)(16,31,32,24)(17,34,21,35)(19,37,33,38)(23,40,27,25)

Polycyclic presentation:

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

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, 416338);