Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22271 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,15)(13,16)(14,17)(18,19,21,23)(20,26,27,33)(22,30,28,25)(24,32,31,29);(6,8,9,7)(10,12,15,11)(13,16,14,17)(19,23)(20,27)(25,30)(29,32);(10,13,15,14)(11,16,12,17)(18,20,28,32,21,27,22,29)(19,24,30,33,23,31,25,26);(10,14,15,13)(11,16,12,17)(19,23)(24,31)(25,30)(26,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15259);