#137386

order	512
nilpotency class	4
isoclinism	isoclinic class of #137383
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33517	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=g,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44582);