#128001

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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

1412 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1412 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)(11,19)(12,23,24,16)(13,17)(14,18)(15,21,27,33)(20,31)(22,32)(25,36,37,29)(26,30)(28,35,38,40)(34,39);(1,3)(2,5,6,7)(4,8)(9,11,13,22,14,20,26,34)(10,15,17,28,18,27,30,38)(12,21,25,35,24,33,37,40)(16,19,29,32,23,31,36,39);(11,20)(15,27)(19,31)(21,33)(22,34)(28,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,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, 35197);