#126471

order	512
nilpotency class	4
isoclinism	isoclinic class of #126283
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#31068	C₂
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,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, 33667);