#126726

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33922);