#144747

order	512
nilpotency class	4
isoclinism	isoclinic class of #144718
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,5)(4,6)(7,8)(9,10,16,21)(11,22,24,17)(12,28,32,64)(13,33,37,67)(14,20,15,38)(18,43,47,30)(19,35,49,36)(23,53,54,70)(25,57,56,66)(26,52,27,42)(29,65,61,62)(31,63,39,48)(34,45,69,46)(40,55,50,68)(41,60,51,59)(44,58,71,72);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,16,24,14,26)(10,17,20,42,21,22,38,52)(12,29,31,54,32,61,39,23)(13,25,36,60,37,56,35,59)(18,44,46,40,47,71,45,50)(19,41,33,66,49,51,67,57)(28,62,48,70,64,65,63,53)(30,58,34,68,43,72,69,55);(5,8)(9,12)(10,18)(11,23,24,54)(13,34)(14,31)(15,39)(16,32)(17,40,22,50)(19,48)(20,45)(21,47)(25,58,56,72)(26,29,27,61)(28,33)(30,36)(35,43)(37,69)(38,46)(41,65,51,62)(42,44,52,71)(49,63)(53,66,70,57)(55,59,68,60)(64,67)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,e^c=eg,g^c=gi

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, 51943);