Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #11999 |
| commutator | C2×C16 | C22×C16 |
| Frattini, cosocle | #7329 | C23 |
| socle | C2 | #72227188 |
| prime radical | C23 | #7456 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C16 ◁ C4×M64 ◁ C8×M64 ◁ G | |
| LECS | G ▷ #7329 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(1,2,5,11,7,14,34,44,8,16,35,46,41,12,50,51,9,18,23,48,6,36,37,63,39,15,22,45,4,28,3,20,10,19,24,49,31,52,25,56,38,57,61,64,21,53,26,43,30,17,58,47,40,42,33,60,29,55,62,59,32,13,27,54);(1,3,21,61,39,33,7,5,30,50,4,22,8,25,40,58,10,27,41,35,29,37,31,24,9,26,32,62,38,34,6,23)(2,11,42,63,57,64,28,54,18,48,14,56,55,59,53,51,19,49,36,60,16,46,13,20,17,47,52,44,15,45,12,43);(1,4,29,6,30,41,38,7,10,32,39,40,9,21,8,31)(2,12,15,52,17,13,16,36,19,53,55,14,18,28,57,42)(3,22,37,23,50,35,34,5,27,62,33,58,26,61,25,24)(11,43,45,44,47,20,46,60,49,51,59,56,48,54,64,63)Polycyclic presentation:
2^11|a^2=d,b^2=ehi,c^2=hik,d^2=f,e^2=gk,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cj,d^b=di,d^c=dk,e^a=eg,e^d=ek,f^b=fk,g^a=gi,i^a=ik224 conjugacy classes — view the full character table.
| maximal subgroup of | #72227195 |
| maximal normal subgroup of | #72227195 |
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, 2, F.5*F.8*F.9);;
gap> SetPower(coll, 3, F.8*F.9*F.11);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.11);;
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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*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);