Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8092 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C22×C42×C5 ◁ G | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,10,13,11,8,9,12)(14,15)(16,24)(17,21)(18,22)(19,23)(20,25)(26,29)(27,28);(6,8,11,7)(9,13,10,12)(14,16,18,26,19,27,17,25)(15,20,21,28,23,29,22,24);(6,9,11,10)(7,12,8,13)(14,17,19,18)(15,21,23,22)(16,25,27,26)(20,28,29,24)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1080);