#502174

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

780 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409370);