#144909

order	512
nilpotency class	4
isoclinism	isoclinic class of #144871
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 68 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)(4,7)(6,8)(9,10,15,20)(11,21,25,43)(12,26,31,56)(13,32,28,18)(14,35,36,19)(16,33,39,23)(17,40,42,30)(22,47,27,57)(24,51,52,46)(29,62,60,55)(34,44,61,54)(37,67,69,72)(38,50,66,65)(41,63,64,70)(45,59,71,49)(48,68,58,53);(1,3)(2,6)(4,8)(5,7)(9,11,15,25)(10,16,20,39)(12,27,31,22)(13,33,28,23)(14,24,36,52)(17,37,42,69)(18,43,32,21)(19,38,35,66)(26,53,56,68)(29,48,60,58)(30,49,40,59)(34,65,61,50)(41,71,64,45)(44,51,54,46)(47,62,57,55)(63,67,70,72);(2,3)(5,6)(7,8)(9,12,28,60,15,31,13,29)(10,17,32,64,20,42,18,41)(11,22,33,58,25,27,23,48)(14,30,61,70,36,40,34,63)(16,37,43,71,39,69,21,45)(19,26,54,62,35,56,44,55)(24,49,65,67,52,59,50,72)(38,68,51,57,66,53,46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dg,d^2=i,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,e^c=egi,f^c=fg,h^c=hi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52105);