Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #97550 |
| commutator | C26 | C22×C4 |
| Frattini, cosocle | C27 | C23 |
| prime radical | #6499867 | C2 |
| LCS | G ▷ C26 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C27 ◁ G | |
| LECS | G ▷ C27 ▷ C24 ▷ C2 ▷ C1 | |
7093 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 7093 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^10|a^2=e,c^2=j,b^a=bd,c^a=cf,c^b=cj,d^a=dg,d^c=di,e^b=eg,e^c=eh,f^a=fh,f^b=fi,g^c=gj,h^b=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72216433, #72216490, #72216504, … |
| maximal normal subgroup of | #72216433, #72216490, #72216504, … |
| maximal quotient of | #72236927, #72236992, #72236999, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);