Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #231631 |
| commutator | C4×C8 | C23×C8 |
| Frattini, cosocle | C42×C8 | C24 |
| socle | C2 | #72216429 |
| prime radical | #491465 | C4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,10,7,14,33,12,8,16,27,42,34,15,6,13)(3,17,20,40,22,37,18,39,24,50,21,9,23,36,52,38)(4,25,28,47,31,49,30,41,32,48,29,46,60,11,43,45)(19,53,54,62,56,44,26,35,58,59,55,61,57,64,51,63);(1,3,8,24)(2,9,14,38,16,40,15,39)(4,26)(5,21,27,20)(6,18,33,52)(7,22,34,23)(10,37,12,50,42,36,13,17)(11,44,48,53,49,64,25,59)(19,28)(29,58)(30,56)(31,55)(32,51)(35,47,62,45,63,46,61,41)(43,57)(54,60);(2,10,14,12,16,42,15,13)(3,18,23,20,24,52,22,21)(11,41,25,46,49,45,48,47)(19,51,57,55,58,26,56,54);(1,4,27,28,7,31,6,30,8,32,5,29,34,60,33,43)(2,11,10,41,14,25,12,46,16,49,42,45,15,48,13,47)(3,19,18,51,23,57,20,55,24,58,52,26,22,56,21,54)(9,35,50,64,39,62,37,59,40,63,17,44,38,61,36,53)Polycyclic presentation:
2^11|a^2=e,b^2=efi,c^2=fg,d^2=gik,e^2=h,f^2=i,g^2=jk,h^2=k,i^2=k,j^2=k,b^a=bf,c^a=cfijk,c^b=cg,d^a=dj,d^b=dgi,e^b=ek,e^d=ek,f^a=fi,f^b=fk,g^b=gj,i^a=ik,j^b=jk164 conjugacy classes — view the full character table.
| maximal subgroup of | #72225512, #72227066, #72227105, … |
| maximal normal subgroup of | #72225512, #72227066, #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.5);;
gap> SetPower(coll, 2, F.5*F.6*F.9);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.7*F.9*F.11);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetPower(coll, 8, F.11);;
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*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, 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.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);