Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×C5×#379 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | C23 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| socle | C24×C5 | C24 |
| prime radical | C26×C5 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ G | |
6734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4)(17,18,19,20,21);(6,8)(9,10,11,12)(14,16);(13,14)(15,16);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6757);