#509146

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
prime radical	C₂²×#135	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

802 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)(9,10,14,19,30,17,36,37,15,20,35,39,12,21,29,24)(11,22,25,32,27,34,18,13,26,31,40,28,23,38,33,16);(5,6)(7,8)(9,11,15,26)(10,16,20,13)(12,27,30,23)(14,33,35,18)(17,28,21,32)(19,38,39,34)(22,37,31,24)(25,36,40,29);(2,4)(6,8)(9,12,15,30)(10,17,20,21)(11,23,26,27)(13,28,16,32)(14,29,35,36)(18,25,33,40)(19,37,39,24)(22,34,31,38);(2,4)(9,13,15,16)(10,18,20,33)(11,24,26,37)(12,28,30,32)(14,34,35,38)(17,25,21,40)(19,27,39,23)(22,29,31,36)

Polycyclic presentation:

2^9|a^2=e,b^2=i,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, 416342);