#137385

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	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 554 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,4,6,5,3,8,7)(9,10,14,20,39,42,37,19,16,22,36,53,15,21,13,35)(11,23,26,17,40,44,25,54,28,57,64,46,27,51,38,45)(12,29,32,50,41,52,30,49,34,47,67,18,33,48,31,55)(24,43,60,56,65,72,59,68,62,71,63,66,61,70,58,69);(1,3)(2,5)(4,6)(7,8)(9,11,15,40,16,28,39,27)(10,17,42,45,22,46,21,54)(12,24,41,61,34,62,33,65)(13,26,36,25,37,64,14,38)(18,43,49,72,50,71,55,70)(19,51,53,57,35,44,20,23)(29,66,48,69,47,56,52,68)(30,63,67,59,31,60,32,58);(1,2)(3,5)(4,6)(7,8)(9,12)(10,18)(11,24)(13,32)(14,31)(15,41)(16,34)(17,43)(19,52)(20,47)(21,55)(22,50)(23,56)(25,63)(26,58)(27,65)(28,62)(29,53)(30,36)(33,39)(35,48)(37,67)(38,60)(40,61)(42,49)(44,69)(45,72)(46,71)(51,68)(54,70)(57,66)(59,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 44581);