#128037

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

1378 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)(11,19,20,30)(12,23,25,16)(13,17)(14,18)(15,21)(22,34,35,40)(24,28)(26,33,36,31)(27,32)(29,37)(38,39);(1,3)(2,5,6,7)(4,8)(9,11,13,22,14,20,27,35)(10,15,17,29,18,28,32,39)(12,24,26,38,25,21,36,37)(16,30,31,40,23,19,33,34);(11,20)(15,28)(19,30)(21,24)(22,35)(29,39)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,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, 35233);