Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C16 · Schur multiplier (H2) = C22 |
| direct product | C5×SD16⋊C16 |
| unambiguous semidirect product | SD16⋊(C5×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C4 | C22×C5×C16 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×D4 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C4×C5×C16 ◁ G | |
232 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)(8,10)(9,11)(12,13)(14,15,17,21,18,22,25,28,19,16,20,23,26,24,27,29);(7,10)(8,13)(11,12);(7,11)(10,12)(14,16,17,23,18,24,25,29,19,15,20,21,26,22,27,28)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5043);