#144709

order	512
nilpotency class	4
isoclinism	isoclinic class of #144670
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	#123
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 419 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)(9,10,15,20)(11,21,25,42)(12,26,30,44)(13,31,34,18)(14,19,36,45)(16,32,39,23)(17,35,41,29)(22,48,27,57)(24,47,53,71)(28,59,61,56)(33,55,62,43)(37,66,68,72)(38,64,69,51)(40,60,63,65)(46,52,70,50)(49,67,58,54);(2,5)(6,7)(9,11,15,25)(10,16,20,39)(12,27,30,22)(13,32,34,23)(14,24,36,53)(17,37,41,68)(18,42,31,21)(19,38,45,69)(26,54,44,67)(28,49,61,58)(29,50,35,52)(33,64,62,51)(40,70,63,46)(43,71,55,47)(48,59,57,56)(60,66,65,72);(2,6)(4,8)(5,7)(9,12,13,28,15,30,34,61)(10,17,18,40,20,41,31,63)(11,22,23,49,25,27,32,58)(14,35,33,65,36,29,62,60)(16,37,21,46,39,68,42,70)(19,44,43,59,45,26,55,56)(24,52,51,66,53,50,64,72)(38,54,47,57,69,67,71,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dg,d^2=i,e^2=g,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,c^b=cf,d^b=di,e^b=eh,e^c=egi,f^c=fi,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, 51905);