Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#2308 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | #72217511 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C27 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
At least 7742 conjugacy classes of subgroups; only the 8140 normal subgroups are shown.
Polycyclic presentation:
2^11|b^2=g,d^b=dh,d^c=dj,e^a=eh,e^b=ei,f^a=fj,g^d=gk,h^b=hk,i^a=ik212 conjugacy classes — view the full character table.
| maximal subgroup of | #72309350, #72309383, #72309388, … |
| maximal normal subgroup of | #72309350, #72309383, #72309388, … |
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);