#128056

order	512
nilpotency class	4
isoclinism	isoclinic class of #128055
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1354 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)(9,10,13,18,15,20,14,19)(11,21,23,36,25,38,24,37)(12,26,27,35,28,17,29,33)(16,30,31,22,32,39,34,40);(1,3)(2,5,6,7)(4,8)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,34)(12,22,27,39,28,40,29,30)(17,21,33,37,26,38,35,36);(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,21)(18,35)(19,33)(20,26)(23,30)(24,39)(25,40)(31,37)(32,38)(34,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fh,g^a=gi,d^b=dh,e^b=eg,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, 35252);