Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60308 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#3483 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,19,22,18,17,21,26)(16,23,25,29,24,20,27,28);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,23)(18,24)(19,25)(21,27)(22,28)(26,29);(8,13)(10,12)(14,16)(15,20)(17,23)(18,24)(19,25)(21,27)(22,28)(26,29);(7,11)(10,12)(16,24)(20,23)(25,27)(28,29);(14,17,19,26,18,15,21,22)(16,23,25,29,24,20,27,28)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53296);