#144965

order	512
nilpotency class	4
isoclinism	isoclinic class of #144871
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=dg,d^2=i,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,e^c=egi,f^c=fg,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, 52161);