#72282449

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#117591
commutator	C₂×#131	C₂²×C₄
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=d,c^2=dghi,f^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=ce,d^c=dh,e^a=ej,e^b=eh,e^c=ej,e^d=ej,f^a=fi,f^b=fh,f^c=fi,f^d=fj,g^c=ghj,g^e=gj,g^f=gj,h^a=hj,h^b=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282517
maximal normal subgroup of	#72282517

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);