#126486

order	512
nilpotency class	4
isoclinism	isoclinic class of #125892
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1629
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 296 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=i,d^2=f,e^2=h,f^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=eg,f^b=fi,d^c=dg,e^c=ehi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33682);