#128006

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
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

1344 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gi,c^b=ci,d^b=dh,e^b=eg,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, 35202);