Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#120 |
| commutator | C24 | C24×C4 |
| Frattini, cosocle | C23×C4 | C25 |
| socle | C22 | C22×#386 |
| prime radical | C2×#3449 | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
9457 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^10|b^2=i,e^2=g,g^2=j,d^a=dh,d^c=di,e^a=ei,e^b=ej,e^d=ef,f^a=fi,h^e=hi160 conjugacy classes — view the full character table.
| maximal subgroup of | #72257574 |
| maximal normal subgroup of | #72257574 |
| maximal quotient of | #72264479 |
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.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);