#509137

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

851 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)(9,10,16,23,29,24,28,18,15,11,13,19,14,20,31,22)(12,17,30,36,39,37,38,32,27,21,25,33,26,34,40,35);(5,6)(7,8)(9,11,15,10)(12,21,27,17)(13,22,16,18)(14,20,29,24)(19,28,23,31)(25,35,30,32)(26,34,39,37)(33,38,36,40);(2,4)(6,8)(9,12,15,27)(10,17,11,21)(13,25,16,30)(14,26,29,39)(18,32,22,35)(19,33,23,36)(20,34,24,37)(28,38,31,40);(2,4)(10,18)(11,22)(13,28)(14,29)(16,31)(17,32)(19,20)(21,35)(23,24)(25,38)(26,39)(30,40)(33,34)(36,37)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 416333);