Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #123275 |
| commutator | C8×C16 | C22×C4 |
| Frattini, cosocle | C2×C8×C16 | C23 |
| socle | C2 | #72225198 |
| prime radical | #72216236 | C2 |
| LCS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ G | |
| LECS | G ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,4,11,8,16,7,14)(3,17,18,50,21,53,32,40)(5,24,22,43,28,42,27,10)(6,12,23,44,31,45,30,15)(9,33,35,57,38,62,39,19)(13,46,41,26,49,60,48,54)(20,34,29,52,58,63,55,47)(25,51,59,64,56,36,61,37);(1,3)(2,9)(4,18)(5,25)(6,29)(7,32)(8,21)(10,40)(11,35)(12,37)(13,47)(14,39)(15,36)(16,38)(17,24)(19,54)(20,30)(22,59)(23,58)(26,57)(27,61)(28,56)(31,55)(33,46)(34,41)(42,53)(43,50)(44,51)(45,64)(48,63)(49,52)(60,62);(2,10,15,13,14,42,45,48,16,43,44,49,11,24,12,41)(9,34,37,17,35,52,51,50,38,63,64,53,39,47,36,40)Polycyclic presentation:
2^11|a^2=d,c^2=eg,d^2=k,e^2=gi,f^2=hjk,g^2=i,h^2=jk,j^2=k,b^a=bf,c^a=cfj,c^b=ce,e^a=eh,e^b=eg,f^a=fhk,f^b=fhk,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk188 conjugacy classes — view the full character table.
| maximal subgroup of | #72226858, #72226961, #72226963 |
| maximal normal subgroup of | #72226858, #72226961, #72226963 |
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, 3, F.5*F.7);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*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.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);