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 | #135908 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | C2×#508 | C2×C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ #11405 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,14,9,16,7,15)(3,17,21,53,24,55,22,61)(4,25,28,11,8,36,26,12)(5,29,33,62,35,45,34,63)(10,38,41,18,43,57,50,20)(13,46,47,27,49,31,48,32)(19,54,23,52,37,64,60,39)(30,51,58,40,56,42,59,44);(1,3,4,19,7,22,26,60,9,24,8,37,6,21,28,23)(2,10,12,40,14,41,25,42,16,43,11,44,15,50,36,51)(5,30,32,57,34,59,31,18,35,56,27,38,33,58,46,20)(13,39,45,61,47,54,63,17,49,52,29,53,48,64,62,55);(2,11,16,12)(3,18,21,38,24,20,22,57)(4,26)(5,31,35,46)(6,7)(8,28)(10,39,41,64,43,52,50,54)(14,25,15,36)(17,51,61,40,55,42,53,44)(19,59,23,56,37,58,60,30)(27,34,32,33)(29,62)(45,63)(47,48)Polycyclic presentation:
2^10|a^2=f,b^2=d,c^2=dhij,d^2=g,e^2=hi,f^2=j,g^2=j,h^2=ij,i^2=j,b^a=be,c^a=cfh,c^b=cehj,d^a=dij,d^c=dij,e^a=eh,e^b=ehij,e^c=ehij,e^d=ej,f^c=fj,g^a=gj,g^c=gj,h^a=hi,h^b=hij,h^c=hij,i^a=ij,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72227053, #72227147, #72227151, … |
| maximal normal subgroup of | #72227053, #72227147, #72227151, … |
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.8*F.9*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
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);