Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #8145 |
| commutator | C2×C8 | C2×C4×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| socle | C2 | #128259 |
| prime radical | C2×#41 | #97 |
| LCS | G ▷ C2×C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
479 conjugacy classes of subgroups.
Polycyclic presentation:
2^10|a^2=f,b^2=d,c^2=g,d^2=g,f^2=h,h^2=i,i^2=j,b^a=bhij,c^a=ce,d^a=di,e^c=ej,f^b=fi,f^d=fj,g^a=gj,h^b=hj130 conjugacy classes — view the full character table.
| maximal subgroup of | #72217249, #72217250, #72225088, … |
| maximal normal subgroup of | #72217249, #72217250, #72225088, … |
| maximal quotient of | C2×#72217203 |
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> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);