#144752

order	512
nilpotency class	4
isoclinism	isoclinic class of #144716
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

376 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=egi,g^c=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51948);