Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#10706 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | D4 |
| commutator | C4 | C43×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C23×C5 | C4×D4 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ C23×C5×C8 ◁ G | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12,13)(14,15)(16,21)(17,20)(18,22)(19,23)(24,27)(25,28)(26,29);(7,9)(15,20,23,22)(21,27,29,28);(14,16,18,25,19,26,17,24)(15,21,22,28,23,29,20,27)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3694);