Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #141341 |
| commutator | C4×C16 | C22×C8 |
| Frattini, cosocle | C4×#346 | C23 |
| socle | C2 | #72216299 |
| prime radical | #491465 | C4 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ C8×#346 ◁ G | |
| LECS | G ▷ C4×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11,7,14,25,43,9,16,8,32,26,15,6,13)(3,17,20,49,22,50,58,51,23,44,33,63,56,52,19,55)(5,27,28,35,30,42,29,47,31,62,24,12,18,45,54,46)(10,34,36,53,38,64,48,21,40,59,37,60,41,57,39,61);(1,3,18,53,6,19,28,59,26,56,31,61,8,33,54,64,9,23,30,60,25,58,24,34,7,22,5,21,4,20,29,57)(2,10,35,17,32,36,42,63,14,38,47,50,13,48,62,55,16,40,12,44,11,37,45,49,15,41,46,52,43,39,27,51);(2,11,14,43,16,32,15,13)(3,19,56,33,23,58,22,20)(12,42,46,62,35,45,47,27)(21,57,53,59,61,64,60,34)Polycyclic presentation:
2^11|a^2=d,b^2=dehik,c^2=fhk,d^2=g,e^2=hijk,f^2=ij,g^2=k,h^2=j,j^2=k,b^a=be,c^a=chi,c^b=cf,d^b=di,e^a=eh,e^b=ei,e^d=ek,f^b=fijk,h^a=hj,h^b=hk,i^a=ik,i^b=ik,j^a=jk140 conjugacy classes — view the full character table.
| maximal subgroup of | #72225374, #72227091, #72227105, … |
| maximal normal subgroup of | #72225374, #72227091, #72227105, … |
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.4);;
gap> SetPower(coll, 2, F.4*F.5*F.8*F.9*F.11);;
gap> SetPower(coll, 3, F.6*F.8*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
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.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);