#144716

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	#29736	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 459 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=dg,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=egi,f^c=fi,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, 51912);