Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#16601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1750 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)(7,9)(11,13)(14,15,17,19)(16,20,21,18)(22,23,27,25)(24,29,28,26);(14,15)(16,18)(17,19)(20,21)(22,24,27,28)(23,26,25,29);(10,11)(12,13)(14,15)(16,20)(17,19)(18,21)(22,25,27,23)(24,26,28,29);(6,7)(8,9)(10,11)(12,13)(14,16)(15,18)(17,21)(19,20)(22,26)(23,28)(24,25)(27,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 9589);