Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C213×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×#120 |
| commutator, Frattini, cosocle | C26 | C25 |
| LCS | G ▷ C26 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C28 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
2144126 conjugacy classes of subgroups.
Polycyclic presentation:
2^11|b^a=bg,c^a=cf,d^b=dh,d^c=dhi,e^b=ehi,e^c=ej,f^b=fk,g^c=gk212 conjugacy classes — view the full character table.
| maximal subgroup of | #72334284, #72338094 |
| maximal normal subgroup of | #72224447, #72334284, #72338094 |
| maximal quotient of | #72267008, #72267040, #72267324, … |
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> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);