#509124

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	#13304	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

833 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)(9,10,12,18,26,33,38,29,13,16,27,31,14,19,24,15)(11,20,21,35,28,36,40,17,22,30,39,34,23,37,25,32);(5,6)(7,8)(10,15)(11,21)(12,24)(14,26)(16,29)(17,34)(18,19)(22,39)(23,40)(25,28)(27,38)(31,33)(32,35)(36,37);(2,4)(6,8)(10,16)(15,29)(17,32)(18,31)(19,33)(20,30)(34,35)(36,37);(2,4)(9,11,13,22)(10,17,16,32)(12,25,27,40)(14,28,26,23)(15,30,29,20)(18,36,31,37)(19,34,33,35)(21,38,39,24)

Polycyclic presentation:

2^9|a^2=e,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, 416320);