#509143

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	C₂³×D₈	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

865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(9,10,12,18,26,33,39,31,13,16,27,32,14,19,25,17)(11,20,23,35,28,40,37,29,21,36,38,30,24,34,22,15);(5,6)(7,8)(9,11,13,21)(10,15,16,29)(12,22,27,37)(14,28,26,24)(17,20,31,36)(18,34,32,40)(19,35,33,30)(23,39,38,25);(2,4)(6,8)(10,16)(11,21)(17,31)(18,32)(19,33)(22,37)(23,38)(24,28);(2,4)(10,17)(11,22)(12,25)(14,26)(16,31)(18,19)(20,34)(21,37)(23,24)(27,39)(28,38)(30,35)(32,33)(36,40)

Polycyclic presentation:

2^9|a^2=e,b^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, 416339);