Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #7385 |
| commutator | C32 | C2×C32 |
| Frattini, cosocle | #93664 | C22 |
| socle | C2 | #72226997 |
| prime radical | #2262 | C16 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ C4×C32 ◁ C8×C32 ◁ #93664 ◁ G | |
| LECS | G ▷ #93664 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 64 points:
(1,2,7,15)(3,8,18,33)(4,9,22,37)(5,11,26,45)(6,13,27,16)(10,39,43,57)(12,46,47,56)(14,25,44,55)(17,31,49,34)(19,38,24,36)(20,42,52,28)(21,29,53,61)(23,35,58,48)(30,62,41,63)(32,51,60,64)(40,54,59,50);(2,8,9,28,11,29,35,43,13,31,36,59,44,60,47,30,15,33,37,42,45,61,48,10,16,34,38,40,14,32,12,41)(3,17,18,49)(4,22)(19,24)(20,54,52,50)(21,51,53,64)(23,58)(39,62,57,63)(46,56)Polycyclic presentation:
2^11|a^2=k,b^2=c,c^2=e,d^2=fjk,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bd,c^a=cf,d^a=dfh,d^b=dj,d^c=dk,e^a=eh,f^a=fhjk,f^b=fk,g^a=gj,h^a=hjk,i^a=ik,j^a=jk224 conjugacy classes — view the full character table.
| maximal subgroup of | #72227001 |
| maximal normal subgroup of | #72227001 |
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.11);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.10*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);