Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #106654 |
| commutator | C23×C4 | C2×C42 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | C2×#3500 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
2997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,14)(3,15,20,31)(4,10,24,37)(5,13,7,11)(6,12,28,41)(9,17,33,19)(16,45,47,43)(18,44,50,59)(21,52,55,61)(22,42,29,35)(23,36,26,39)(25,40,27,38)(30,51,58,46)(32,48,60,49)(34,53,62,54)(56,64,57,63);(1,3)(2,9)(4,21)(5,17,7,19)(6,18)(8,20)(10,34)(11,31,13,15)(12,32)(14,33)(16,23)(22,54,29,53)(24,55)(25,48,27,49)(26,47)(28,50)(30,36)(35,52,42,61)(37,62)(38,59,40,44)(39,58)(41,60)(43,63,45,64)(46,57,51,56);(1,4,5,22,8,24,7,29)(2,10,11,35,14,37,13,42)(3,16,17,46,20,47,19,51)(6,26,27,56,28,23,25,57)(9,30,31,45,33,58,15,43)(12,39,40,63,41,36,38,64)(18,21,49,53,50,55,48,54)(32,34,44,61,60,62,59,52)Polycyclic presentation:
2^10|a^2=j,b^2=g,c^2=d,d^2=j,e^2=j,f^2=j,b^a=bf,c^a=ch,c^b=ce,d^a=dj,d^b=dh,e^b=ei,e^c=ehj,e^d=ej,f^a=fj,f^b=fhi,f^c=fj,g^a=ghij,g^c=gij,g^d=gj,g^f=gj,h^b=hj,h^c=hj,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72311163, #72311270, #72311394, … |
| maximal normal subgroup of | #72311163, #72311270, #72311394, … |
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.7);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);