#144855

order	512
nilpotency class	4
isoclinism	isoclinic class of #144670
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	#123
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 335 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 52051);