Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#120 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C26 | C24 |
| prime radical | C24×#120 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C27 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
148069 conjugacy classes of subgroups.
Polycyclic presentation:
2^10|a^2=f,b^a=be,c^a=cg,d^a=di,d^b=dh,d^c=dj,e^d=ej,h^a=hj160 conjugacy classes — view the full character table.
| maximal subgroup of | #72218522, #72218799, #72218856, … |
| maximal normal subgroup of | #72218477, #72218522, #72218799, … |
| maximal quotient of | #72225818, #72238520, #72250356, … |
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.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);