#72250061

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346522
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=hj,d^2=efgi,e^2=j,f^2=i,g^2=j,i^2=j,c^a=cj,d^a=de,d^b=df,d^c=degj,e^a=ej,e^c=eh,e^d=eh,f^b=fi,f^c=fj,g^a=gh,g^b=gj,g^c=gh,g^d=gh,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250237
maximal normal subgroup of	#72250237
maximal quotient of	#72250637

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, 1, F.10);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);