#126496

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	C₂³	#453
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

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

normal subgroups only

This group has 268 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,28,31,46)(20,32)(21,34)(22,26)(23,38)(24,29,42,48)(25,33)(27,37)(30,40,39,45)(35,43,47,41)(36,44);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,31)(18,24,26,42)(20,29,33,48)(21,35,36,47)(23,39,37,30)(25,40,32,45)(27,43,38,41)(28,34,46,44);(17,20,22,33)(18,25,26,32)(19,29,31,48)(21,23,36,37)(24,40,42,45)(27,44,38,34)(28,41,46,43)(30,35,39,47)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=i,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=dfi,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi

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, 33692);