#128076

order	512
nilpotency class	4
isoclinism	isoclinic class of #128057
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

1388 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)(11,21,23,32)(12,18,13,17)(14,19)(15,20)(16,22)(24,35,37,40)(25,33,27,31)(26,30)(28,34)(29,36)(38,39);(1,3)(2,5,6,7)(4,8)(9,11,14,24,15,23,28,37)(10,16,19,29,20,30,34,39)(12,22,25,36,13,26,27,38)(17,21,31,35,18,32,33,40);(9,12,15,13)(10,17,20,18)(11,22,23,26)(14,25,28,27)(16,21,30,32)(19,31,34,33)(24,36,37,38)(29,35,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35272);