Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23×D4 |
| commutator | C22×C4 | C24×C4 |
| Frattini, cosocle | C23×C4 | C25 |
| prime radical | C22×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
10495 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
2^10|a^2=h,c^2=h,d^2=f,e^2=i,i^2=j,d^b=dh,e^a=ei,e^b=eg,e^c=ej,i^a=ij172 conjugacy classes — view the full character table.
| maximal subgroup of | #72249833, #72305845, #72306198, … |
| maximal normal subgroup of | #72249833, #72305845, #72306198, … |
| maximal quotient of | #72250849, #72322784 |
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.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);