Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #122887 |
| commutator | C82 | C22×C8 |
| Frattini, cosocle | C2×C8×C16 | C23 |
| socle | C2 | #72216566 |
| prime radical | #491465 | C4 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ C8×#346 ◁ G | |
| LECS | G ▷ C2×C8×C16 ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
858 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 858 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(1,2,4,11,5,13,26,43,8,16,29,46,6,12,27,42,9,17,30,47,7,14,28,44,32,15,50,45,33,48,59,64)(3,18,20,51,21,10,25,36,23,37,19,41,31,39,53,35,24,49,54,60,57,40,58,61,34,52,55,63,22,38,56,62);(1,3)(2,10)(4,25)(5,31)(6,21)(7,22)(8,34)(9,24)(11,41)(12,37)(13,49)(14,18)(15,39)(16,38)(17,40)(19,27)(20,28)(23,32)(26,54)(29,56)(30,58)(33,57)(35,42)(36,44)(43,61)(45,60)(46,51)(47,63)(48,52)(50,53)(55,59)(62,64);(1,4,7,28,8,29,33,59,9,30,5,26,32,50,6,27)(2,11,14,44,16,46,48,64,17,47,13,43,15,45,12,42)(3,19,21,53,23,54,31,58,24,55,57,56,34,20,22,25)(10,35,37,60,39,61,49,63,40,62,52,51,38,36,18,41)Polycyclic presentation:
2^11|a^2=d,c^2=h,d^2=f,e^2=ghijk,f^2=j,g^2=ik,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cij,c^b=chjk,d^b=dh,d^c=dk,e^a=eg,e^b=egh,f^b=fj,g^a=gi,g^b=gi,h^b=hjk,i^a=ik,i^b=ik,j^b=jk140 conjugacy classes — view the full character table.
| maximal subgroup of | #72227228, #72227232, #72227242, … |
| maximal normal subgroup of | #72227228, #72227232, #72227242, … |
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.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);