#126484

order	512
nilpotency class	4
isoclinism	isoclinic class of #126351
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,5,8,6)(9,10,12,15,14,19,26,17)(11,20,21,35,23,38,24,37)(13,27,25,18,30,32,29,36)(16,28,33,22,34,39,31,40);(2,5)(3,8)(6,7)(10,15,19,17)(16,31,34,33)(18,27,36,32)(20,35,38,37);(9,11,14,23)(10,16,19,34)(12,24,26,21)(13,28,30,39)(15,31,17,33)(18,35,36,37)(20,27,38,32)(22,25,40,29)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=i,d^2=f,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33680);