#127050

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34246);