Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22267 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C5×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
994 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)(7,9)(10,11)(12,15)(13,16)(14,17)(18,19,23,25,22,26,21,30)(20,24,28,33,29,27,31,32);(10,12,15,11)(13,16,14,17)(18,19,21,30,22,26,23,25)(20,27,31,33,29,24,28,32);(6,7,8,9)(10,13,15,14)(11,16,12,17)(18,20,22,29)(19,24,26,27)(21,31,23,28)(25,33,30,32);(10,14,15,13)(11,16,12,17)(18,21,22,23)(19,25,26,30)(20,28,29,31)(24,32,27,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15255);