Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10858 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(11,13)(14,15,18,22,17,21,19,26)(16,20,24,28,23,27,25,29);(7,9)(14,16,19,24,17,23,18,25)(15,20,22,29,21,27,26,28);(10,11,12,13)(15,21)(20,27)(22,26)(28,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3846);