#128038

order	512
nilpotency class	4
isoclinism	isoclinic class of #128009
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1364 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)(4,6)(5,8)(9,10)(11,21,24,35)(12,17)(13,27,29,18)(14,19)(15,20)(16,23)(22,37,25,39)(26,36,30,34)(28,32)(31,38)(33,40);(1,3)(2,5,6,7)(4,8)(9,11,14,22,15,24,12,25)(10,16,19,31,20,32,17,33)(13,28,30,40,29,23,26,38)(18,35,36,39,27,21,34,37);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,29,30)(16,31,32,33)(18,34,27,36)(21,37,35,39)(23,38,28,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gi,c^b=ci,d^b=dh,e^b=egi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35234);