#144779

order	512
nilpotency class	4
isoclinism	isoclinic class of #144686
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 339 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)(12,28,30,63)(13,33,38,67)(14,20,15,39)(18,44,46,31)(19,36,51,37)(23,54,56,70)(25,57,34,68)(26,53,27,43)(29,62,61,72)(32,50,40,64)(35,48,66,47)(41,55,52,69)(42,60,49,59)(45,65,71,58);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,16,24,14,26)(10,17,20,43,21,22,39,53)(12,29,32,56,30,61,40,23)(13,34,37,59,38,25,36,60)(18,45,48,41,46,71,47,52)(19,49,33,57,51,42,67,68)(28,62,64,54,63,72,50,70)(31,58,66,55,44,65,35,69);(5,8)(9,12,15,40,16,30,14,32)(10,18,20,47,21,46,39,48)(11,23,26,29,24,56,27,61)(13,35,37,44,38,66,36,31)(17,41,53,71,22,52,43,45)(19,50,33,63,51,64,67,28)(25,58,59,55,34,65,60,69)(42,72,57,70,49,62,68,54)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dg,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,d^c=di,e^c=egi,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, 51975);