#509130

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
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

822 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)(9,10,14,19,12,25,26,31,15,13,27,29,16,18,32,36)(11,20,23,34,33,22,39,28,24,38,40,35,21,37,30,17);(5,6)(7,8)(9,11,15,24)(10,17,13,28)(12,21,16,33)(14,30,27,39)(18,34,25,35)(19,37,29,22)(20,31,38,36)(23,26,40,32);(2,4)(6,8)(9,12,15,16)(10,18,13,25)(11,21,24,33)(14,26,27,32)(17,34,28,35)(19,36,29,31)(20,22,38,37)(23,30,40,39);(2,4)(9,13,15,10)(11,22,24,37)(12,25,16,18)(14,31,27,36)(17,30,28,39)(19,32,29,26)(20,21,38,33)(23,34,40,35)

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=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, 416326);