#144873

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

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

normal subgroups only

This group has 420 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)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,25,43)(12,26,30,55)(13,31,34,18)(14,35,36,19)(16,32,39,23)(17,40,42,29)(22,47,27,56)(24,51,52,46)(28,59,61,54)(33,44,65,63)(37,67,69,72)(38,50,66,64)(41,60,62,70)(45,58,71,49)(48,68,57,53);(1,2)(3,7)(4,6)(5,8)(9,11,15,25)(10,16,20,39)(12,27,30,22)(13,32,34,23)(14,24,36,52)(17,37,42,69)(18,43,31,21)(19,38,35,66)(26,53,55,68)(28,48,61,57)(29,49,40,58)(33,64,65,50)(41,71,62,45)(44,51,63,46)(47,59,56,54)(60,67,70,72);(2,6)(3,4)(7,8)(9,12,13,28,15,30,34,61)(10,17,18,41,20,42,31,62)(11,22,23,48,25,27,32,57)(14,29,33,60,36,40,65,70)(16,37,21,45,39,69,43,71)(19,26,44,54,35,55,63,59)(24,49,50,72,52,58,64,67)(38,68,46,47,66,53,51,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,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, 52069);