Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24246 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#548 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,8)(10,11)(12,13)(14,15)(16,24)(17,21)(18,22)(19,23)(20,27)(25,28)(26,29);(6,7,9,8)(14,15)(16,25)(17,21)(18,22)(19,23)(20,26)(24,28)(27,29);(11,13)(14,16,19,28)(15,20,23,29)(17,26,18,27)(21,24,22,25);(6,8,9,7)(14,17,19,18)(15,21,23,22)(16,26,28,27)(20,24,29,25)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17234);