Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#34261 |
| 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 | |
2718 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)(6,7,9,8)(10,11,14,15)(12,13,16,17)(18,19)(20,25)(21,24)(22,23);(11,15)(12,16)(19,23)(24,25);(6,8,9,7)(18,20,22,21)(19,24,23,25);(10,12)(11,13)(14,16)(15,17)(18,19)(20,24)(21,25)(22,23);(10,13)(11,12)(14,17)(15,16)(18,21,22,20)(19,25,23,24)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27249);