Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #1384 |
| commutator | C2×C16 | C4×C16 |
| Frattini, cosocle | C8×M64 | C22 |
| socle | C2 | #72227190 |
| prime radical | C23 | #7459 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C4×C16 ◁ C4×M64 ◁ C8×M64 ◁ G | |
| LECS | G ▷ C8×M64 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,3,12,5,15,20,43,7,16,22,44,32,52,55,64,9,17,24,45,29,14,50,51,6,33,21,57,31,13,47,48,10,18,19,46,4,11,25,36,26,38,54,61,28,39,60,62,8,35,23,58,30,40,56,42,34,37,53,63,27,49,59,41);(2,11,16,39,17,40,33,49,18,15,38,52,35,14,37,13)(3,19)(12,36,44,62,45,42,57,41,46,43,61,64,58,51,63,48)(20,25)(21,53)(22,54)(23,24)(47,59)(50,56)(55,60)Polycyclic presentation:
2^11|a^2=c,b^2=dehi,c^2=f,d^2=gjk,e^2=hij,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bd,c^b=cijk,d^a=dg,d^c=dk,e^a=ej,e^c=ek,g^a=gi,i^a=ik272 conjugacy classes — view the full character table.
| maximal subgroup of | #72227195, #72227272, #72227273 |
| maximal normal subgroup of | #72227195, #72227272, #72227273 |
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.3);;
gap> SetPower(coll, 2, F.4*F.5*F.8*F.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.10*F.11);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
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, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);