#509597

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

685 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)(9,10,15,20)(11,21,25,39)(12,27,18,17)(13,29,19,37)(14,31,33,28)(16,34,35,24)(22,38,26,30)(23,40,36,32);(5,6)(7,8)(9,11,15,25)(10,16,20,35)(12,22,18,26)(13,30,19,38)(14,32,33,40)(17,23,27,36)(21,28,39,31)(24,37,34,29);(2,4)(6,8)(9,12,15,18)(10,17,20,27)(11,22,25,26)(13,28,19,31)(14,29,33,37)(16,23,35,36)(21,30,39,38)(24,40,34,32);(2,4)(9,13,29,20,12,28,33,27,15,19,37,10,18,31,14,17)(11,23,32,21,26,16,34,38,25,36,40,39,22,35,24,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=eh,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, 416793);