Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×#392 |
| commutator | C26 | C23×C4 |
| Frattini, cosocle | C25×C4 | C24 |
| socle | C23 | C2×#1438 |
| prime radical | C2×#34736 | C4 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C42 ◁ G | |
| LECS | G ▷ C25×C4 ▷ C23 ▷ C1 | |
At least 6903 conjugacy classes of subgroups; only the 876 normal subgroups are shown.
Polycyclic presentation:
2^11|b^2=j,d^2=f,f^2=k,c^a=ch,d^a=dg,d^b=dk,d^c=de,e^a=ei,g^c=gj,h^d=hij212 conjugacy classes — view the full character table.
| maximal subgroup of | #72265091 |
| maximal normal subgroup of | #72265091 |
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, 2, F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);