Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C4×#442 |
| commutator | C2×C8 | C22×C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C24 |
| socle | C2 | #72225524 |
| prime radical | C2×#508 | C2×C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#346 ◁ C4×#1398 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,9,16)(3,17,24,39)(4,25,30,47)(5,31,29,12)(6,11,34,14)(7,35,8,15)(10,37,42,21)(13,27,44,32)(18,41,22,40)(19,54,26,61)(20,57,58,49)(23,48,36,50)(28,45,62,60)(33,46,51,43)(38,52,63,56)(53,59,55,64);(1,3,5,20,6,22,33,55,8,23,4,26,27,21,28,52,9,24,29,58,34,18,51,53,7,36,30,19,32,37,62,56)(2,10,12,38,11,39,43,57,15,41,47,64,44,50,60,61,16,42,31,63,14,17,46,49,35,40,25,59,13,48,45,54);(2,11,15,44,16,14,35,13)(3,18,23,37,24,22,36,21)(12,43,47,60,31,46,25,45)(19,52,20,53,26,56,58,55);(1,4,6,28,8,29,27,51,9,30,34,62,7,5,32,33)(2,12,14,46,15,47,13,45,16,31,11,43,35,25,44,60)(3,19,22,56,23,20,21,55,24,26,18,52,36,58,37,53)(10,38,17,49,41,64,48,54,42,63,39,57,40,59,50,61)Polycyclic presentation:
2^11|a^2=k,b^2=e,c^2=fgjk,d^2=g,e^2=h,f^2=ik,g^2=j,h^2=j,i^2=k,j^2=k,b^a=bf,c^a=cf,c^b=cfijk,d^a=dik,d^b=dij,e^a=ek,e^d=ek,f^a=fi,f^b=fik,g^a=gk,i^a=ik,i^b=ik248 conjugacy classes — view the full character table.
| maximal subgroup of | #72225554, #72227037, #72227091, … |
| maximal normal subgroup of | #72225554, #72227037, #72227091, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.7*F.10*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);