Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72224639 |
| commutator | C32 | C22×C16 |
| Frattini, cosocle | #7330 | C23 |
| prime radical | #13742 | C8 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1311 ◁ #93668 ◁ G | |
| LECS | G ▷ #7330 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
218 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,3,8,4,9,13,26,5,10,14,28,17,34,40,23,6,11,15,29,18,32,41,52,20,30,43,24,47,54,37,53,7,12,16,27,19,33,39,59,21,35,44,56,46,31,62,51,22,36,42,25,48,57,38,58,49,60,64,55,45,61,63,50);(2,8,10,24,11,25,30,28,12,27,35,55,36,29,60,56)(3,13,15,38,16,39,42,41)(4,17,19,46)(5,20)(6,22)(9,23,31,26,32,50,61,58,33,51,34,59,57,53,54,52)(14,37,43,40,44,63,64,62)(18,45,48,47)(21,49);(2,9,30,61,36,57,10,31,12,33,60,54,11,32,35,34)(3,14,42,64,16,44,15,43)(4,18,19,48)(5,21)(8,23,28,58,29,53,24,26,27,51,56,52,25,50,55,59)(13,37,41,62,39,63,38,40)(17,45,46,47)(20,49)Polycyclic presentation:
2^11|a^2=e,b^2=d,c^2=df,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=begijk,c^a=cgijk,d^a=dijk,e^b=eg,e^c=ei,e^d=ej,f^a=fjk,f^e=fk,g^b=gi,g^c=gj,g^d=gk,h^a=hk,i^b=ij,i^c=ik,j^b=jk95 conjugacy classes — view the full character table.
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.4);;
gap> SetPower(coll, 3, F.4*F.6);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);