Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #135826 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | C2×#41 | #106 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ #33326 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11)(3,15,17,37,20,51,34,52)(5,26)(6,10,23,36)(7,31,24,13)(8,33,25,14)(9,35,38,54,40,18,48,57)(12,43)(16,53,27,41,55,39,56,47)(19,59,21,50,32,44,58,64)(22,60)(28,42)(29,45)(30,46)(49,61)(62,63);(1,3,7,32,8,34,6,21,4,20,24,19,25,17,23,58)(2,9,36,53,14,48,13,47,11,40,10,39,33,38,31,41)(5,27,43,18,30,55,61,57,22,56,28,35,62,16,29,54)(12,44,26,51,45,64,63,52,42,59,60,15,49,50,46,37);(2,10,11,36)(3,16,17,27,20,55,34,56)(5,28,22,43)(6,7)(8,25)(9,37,48,15,40,52,38,51)(12,45)(13,33,31,14)(18,21,57,32,35,58,54,19)(23,24)(26,60)(29,30,61,62)(39,64,41,44,53,50,47,59)(42,49)Polycyclic presentation:
2^10|a^2=d,b^2=dfi,c^2=dfh,d^2=g,e^2=hj,f^2=ij,h^2=i,i^2=j,b^a=be,c^a=cf,c^b=ce,d^b=di,e^a=eh,e^b=eh,e^c=eh,e^d=ej,f^a=fi,f^b=fi,f^c=fij,g^b=gj,h^a=hi,h^b=hi,h^c=hi,i^a=ij,i^b=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72227117, #72227145, #72227147, … |
| maximal normal subgroup of | #72227117, #72227145, #72227147, … |
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.4);;
gap> SetPower(coll, 2, F.4*F.6*F.9);;
gap> SetPower(coll, 3, F.4*F.6*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, 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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
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.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);