Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#11599 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C22×C5×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
450 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)(6,7,8,10,9,11,12,13)(14,15,16,20,18,23,24,22)(17,26,25,21,28,19,27,29);(15,19)(16,24)(17,27)(20,21)(22,29)(23,26)(25,28);(14,16,18,24)(15,20,23,22)(17,25,28,27)(19,29,26,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4587);