#501232

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1169 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)(9,10,14,19)(11,20,21,31)(12,16,27,35)(13,28,26,18)(15,30,33,22)(17,36,34,25)(23,40,29,37)(24,38,39,32);(5,6)(7,8)(9,11,14,21)(10,15,19,33)(12,24,27,39)(13,29,26,23)(16,20,35,31)(17,37,34,40)(18,38,28,32)(22,36,30,25);(2,4)(6,8);(10,16)(11,21)(12,25)(13,26)(15,31)(17,18)(19,35)(20,33)(22,39)(24,30)(27,36)(28,34)(32,37)(38,40)

Polycyclic presentation:

2^9|a^2=fi,b^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, 408428);