Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,10,12)(8,11,9,13)(14,15,20,24)(16,26,22,28)(17,27,18,21)(19,25,29,23);(6,8,10,9)(7,11,12,13)(14,16)(15,18)(17,24)(19,21)(20,22)(23,28)(25,26)(27,29);(6,9,10,8)(7,11,12,13)(14,17,20,18)(15,21,24,27)(16,25,22,23)(19,28,29,26);(14,18,20,17)(15,22,24,16)(19,28,29,26)(21,23,27,25)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19393);