Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #140505 |
| commutator | C4×C16 | C22×C8 |
| Frattini, cosocle | C4×#346 | C23 |
| socle | C2 | #72216306 |
| prime radical | C23 | #13188 |
| 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 | |
475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11,7,14,28,10,8,16,6,13,27,17,34,15)(3,18,21,51,24,53,23,52,25,54,20,56,57,45,58,64)(5,29,31,48,32,43,55,49,33,36,26,12,44,46,19,47)(9,35,37,61,39,63,42,62,41,59,50,30,40,60,38,22);(1,3,19,35,4,20,33,61,7,24,31,63,28,58,44,62,8,25,55,59,6,21,5,30,27,57,26,60,34,23,32,22)(2,9,36,56,10,42,49,54,17,40,43,52,11,37,48,53,16,41,29,51,15,38,47,18,14,39,46,64,13,50,12,45);(2,10,17,11,16,15,14,13)(3,20,24,58,25,21,57,23)(12,36,49,43,48,29,47,46)(22,35,61,63,62,59,30,60)Polycyclic presentation:
2^11|a^2=d,b^2=efijk,c^2=fhjk,d^2=g,e^2=hijk,f^2=ij,g^2=k,h^2=j,j^2=k,b^a=be,c^a=chijk,c^b=cfik,d^b=di,e^a=eh,e^b=eij,e^d=ek,f^b=fij,h^a=hj,i^a=ik,i^b=ik,j^a=jk140 conjugacy classes — view the full character table.
| maximal subgroup of | #72225374, #72227077, #72227121, … |
| maximal normal subgroup of | #72225374, #72227077, #72227121, … |
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.5*F.6*F.9*F.10*F.11);;
gap> SetPower(coll, 3, F.6*F.8*F.10*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*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.11);;
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*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
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);