Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator | C4 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C2×C5×#585 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×#41 ◁ G | |
4902 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)(11,16)(13,14)(17,23)(18,22)(20,21)(24,25);(10,11)(12,17)(13,16)(14,22)(15,18)(19,24)(20,23)(21,25);(10,12)(11,17)(13,20)(14,21)(15,19)(16,23)(18,24)(22,25);(12,19)(17,24)(20,21)(23,25);(10,13,15,14)(11,16,18,22)(12,20,19,21)(17,23,24,25)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53077);