Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #154670 |
| commutator | C26 | C23×C4 |
| Frattini, cosocle | C27 | C24 |
| prime radical | C22×#23900 | C2 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C27 ◁ G | |
| LECS | G ▷ C27 ▷ C22 ▷ C1 | |
At least 8352 conjugacy classes of subgroups; only the 964 normal subgroups are shown.
Polycyclic presentation:
2^11|a^2=g,d^2=ik,c^a=ch,c^b=ce,d^a=dgi,d^b=df,d^c=dhj,e^d=ej,f^c=fj,g^c=gj,g^d=gk,h^a=hj,i^a=ik152 conjugacy classes — view the full character table.
| maximal subgroup of | #72331363, #72331812, #72333265, … |
| maximal normal subgroup of | #72331363, #72331812, #72333265, … |
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.7);;
gap> SetPower(coll, 4, F.9*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);