Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(10,11,15,22,21,25,12,18,14,16,23,17,13,19,20,24);(6,7)(8,9)(11,16)(12,20)(15,23)(19,25);(7,9)(11,17,16,22)(12,15)(13,21)(18,19,24,25)(20,23)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6489);