Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#10705 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C4 | C43×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C23×C5 | C2×C4⋊C4 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
870 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)(14,15,17,22,19,23,18,21)(16,24,26,20,25,29,27,28);(7,9)(14,16,18,27,19,25,17,26)(15,20,22,29,23,28,21,24);(10,11,12,13)(16,25)(20,28)(24,29)(26,27)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3693);