Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13303 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| prime radical | C23×C5 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×Q8 ◁ G | |
396 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)(8,12)(9,10)(11,13)(14,15,19,23,20,24,17,21)(16,25,27,18,22,26,29,28);(7,10)(11,12)(14,16,26,23,20,22,25,21)(15,19,29,18,24,17,27,28);(6,8,9,13)(7,11,10,12)(14,17,20,19)(15,21,24,23)(16,27,22,29)(18,26,28,25)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6291);