#144707

order	512
nilpotency class	4
isoclinism	isoclinic class of #144670
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 444 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)(10,15)(11,19)(12,23)(16,31)(17,30)(18,24)(20,36)(21,38)(22,27)(25,34)(26,40)(29,32)(33,37)(35,39);(2,5)(6,7)(9,10,14,18)(11,20,22,16)(12,24,23,15)(13,26,28,17)(19,33,27,32)(21,29,39,37)(25,30,34,40)(31,38,36,35);(2,6)(4,8)(5,7)(9,11,12,21,14,22,23,39)(10,16,15,29,18,20,24,37)(13,27,25,38,28,19,34,35)(17,33,30,36,26,32,40,31)

Polycyclic presentation:

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

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, 51903);