Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C4×C5×#560 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C2×C5×#560 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(5,6)(7,14)(8,13)(9,11)(10,16)(12,18)(15,19)(17,20);(5,7)(6,10)(8,16)(9,17)(11,19)(12,20)(13,14)(15,18);(6,11)(7,15)(10,14)(12,13)(16,17)(19,20);(6,12)(7,16)(10,19)(11,13)(14,20)(15,17)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27339);