Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12638 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,13,12)(14,15)(16,26)(17,21)(18,23)(19,22)(20,25)(24,28)(27,29);(7,9)(14,16,19,21,17,22,26,15,20,23,29,24,28,27,18,25);(10,12,13,11)(14,17,20,28)(15,21,25,24)(16,27,23,22)(18,19,26,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5626);