Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #12078 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| socle | C2 | #136707 |
| prime radical | #33981 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#1313 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,13,32,44,28,46,37,11,27,47,34,45,4,23,8,14,31,42,35,10,25,43,7,16,30,49,6,15,26,12)(3,17,19,41,60,53,57,51,36,50,63,39,61,62,58,38,22,54,59,9,33,40,29,64,21,55,24,52,20,48,18,56);(1,3)(2,9,11,39,14,41,16,52)(4,24,26,63)(5,29,31,57)(6,33)(7,36)(8,22)(10,38,15,51,44,56,45,64)(12,48,43,53,23,62,46,40)(13,50,49,54,42,55,47,17)(18,27,58,30)(19,25,59,28)(20,35)(21,37)(32,61)(34,60);(2,10,16,45,14,44,11,15)(4,25,26,28)(5,30,31,27)(6,34)(9,38,52,64,41,56,39,51)(12,49,46,13,23,47,43,42)(17,53,55,48,54,40,50,62)(18,57,58,29)(19,24,59,63)(20,61)(32,35)(33,60)Polycyclic presentation:
2^10|a^2=f,b^2=d,c^2=dg,d^2=g,e^2=h,f^2=hij,h^2=i,i^2=j,b^a=be,c^a=cefhj,d^a=di,e^a=ej,e^b=eh,e^c=eij,e^d=ej,f^b=fhi,f^c=fi,f^d=fj,g^a=gj,h^b=hi,h^c=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72225073, #72225094, #72225102, … |
| maximal normal subgroup of | #72225073, #72225094, #72225102, … |
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.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
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.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);