Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C4×#2177 |
| commutator | C2×C16 | C22×C4×C8 |
| Frattini, cosocle | C2×C8×C16 | C24 |
| socle | C2 | #72225392 |
| prime radical | #33981 | C2×C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M64 ◁ C4×#1398 ◁ C8×#1398 ◁ G | |
| LECS | G ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,11)(3,16,19,47)(4,9,23,35)(5,12,27,41)(6,8,28,15)(10,36,37,20)(13,43,42,18)(14,26,31,58)(17,48,22,45)(21,34,53,61)(24,32,49,44)(25,29,57,33)(30,50,40,54)(38,62,63,55)(39,64,60,52)(46,56,59,51);(2,8,11,15)(3,17,19,22)(4,23)(9,33,35,29)(10,37)(12,31,41,14)(13,44,42,32)(18,43)(20,54,36,50)(21,51,53,56)(24,49)(25,57)(30,40)(38,63)(39,60)(52,55,64,62);(2,9,12,32,8,29,31,42,11,35,41,44,15,33,14,13)(10,34,38,45,30,59,60,47,37,61,63,48,40,46,39,16);(1,3,18,52,58,51,57,50,28,17,49,62,27,53,23,36,7,19,43,64,26,56,25,54,6,22,24,55,5,21,4,20)(2,10,9,34,12,38,32,45,8,30,29,59,31,60,42,47,11,37,35,61,41,63,44,48,15,40,33,46,14,39,13,16)Polycyclic presentation:
2^12|a^2=l,b^2=e,c^2=f,d^2=gikl,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bk,c^a=cg,c^b=cj,d^a=dg,d^b=dkl,e^a=el,e^d=el,f^a=fi,g^a=gikl,g^b=gl,h^a=hk,i^a=ikl,j^a=jl,k^a=kl376 conjugacy classes — view the full character table.
| maximal subgroup of | #72225550, #72226982, #72227029, … |
| maximal normal subgroup of | #72225550, #72226982, #72227029, … |
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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.12);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9*F.11*F.12);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);