Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21698 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11,13,12)(14,15,19,23)(16,24,28,38)(17,21,29,35)(18,31,25,22)(20,32,34,27)(26,37,40,45)(30,42,39,36)(33,43,44,41);(14,16,25,32,19,28,18,27)(15,20,31,38,23,34,22,24)(17,26,39,43,29,40,30,41)(21,33,42,45,35,44,36,37);(6,7,8,9)(14,17,19,29)(15,21,23,35)(16,26,28,40)(18,30,25,39)(20,33,34,44)(22,36,31,42)(24,37,38,45)(27,41,32,43);(10,12,13,11)(17,29)(21,35)(26,40)(30,39)(33,44)(36,42)(37,45)(41,43)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14686);