Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #14715 |
| commutator, Frattini, cosocle | C26 | C24 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
22959 conjugacy classes of subgroups.
Polycyclic presentation:
2^10|b^2=g,c^2=e,d^2=j,b^a=bg,c^a=ce,c^b=cf,d^c=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fj,g^c=gi,h^b=hj94 conjugacy classes — view the full character table.
| maximal subgroup of | #72220596, #72220729, #72231481, … |
| maximal normal subgroup of | #72213339, #72213344, #72216662, … |
| maximal quotient of | #72213339, #72213344, #72215603, … |
| automorphism group of | #496, #497 |
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.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);