Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11332 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #389 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,19,16,20,14,31)(12,21,24,33,25,38,23,17)(13,26,28,36,30,35,29,18)(22,39,27,34,41,32,40,37);(6,7,8,9)(10,12,14,23,16,25,15,24)(11,17,19,21,20,33,31,38)(13,27,28,41,30,40,29,22)(18,34,35,39,36,37,26,32);(10,13,14,29,16,30,15,28)(11,18,19,26,20,36,31,35)(12,22,23,40,25,41,24,27)(17,32,21,37,33,39,38,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4320);