Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#34377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,14)(13,16)(15,17)(19,22)(20,24);(11,12)(15,16)(19,22)(23,25);(10,11,14,12)(13,15,17,16)(18,19)(20,23)(21,22)(24,25);(10,12,14,11)(13,16,17,15)(18,20,21,24)(19,23,22,25);(10,13)(11,15)(12,16)(14,17)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27365);