#501012

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	C₂²×#135	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

1025 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)(5,6,7,8);(2,4)(9,10,15,19)(11,20,24,38)(12,17,14,30)(13,26,29,37)(16,31,33,22)(18,25,36,28)(21,35,23,40)(27,34,39,32);(6,8)(9,11,15,24)(10,16,19,33)(12,23,14,21)(13,27,29,39)(17,34,30,32)(18,35,36,40)(20,37,38,26)(22,25,31,28);(2,4)(9,12,15,14)(10,17,19,30)(11,21,24,23)(13,25,29,28)(16,32,33,34)(18,26,36,37)(20,35,38,40)(22,39,31,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 408208);