Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C27 |
| socle | C22 | C25×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
67305 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^10|b^2=hj,c^2=hj,f^2=i,i^2=j,b^a=bh,e^d=ej,f^b=fi,f^c=fj,g^b=gj,i^b=ij232 conjugacy classes — view the full character table.
| maximal subgroup of | #72307155, #72307305, #72308106, … |
| maximal normal subgroup of | #72307155, #72307305, #72308106, … |
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, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);