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×#20768 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,23,20,24,18,34)(16,25,29,36,27,39,28,21)(17,30,32,37,33,44,31,22)(26,40,43,45,41,35,42,38);(7,10)(11,13)(14,16,18,28,20,27,19,29)(15,21,23,25,24,36,34,39)(17,26,31,42,33,41,32,43)(22,35,30,38,37,40,44,45);(6,8,9,12)(7,11,10,13)(14,17,18,31,20,33,19,32)(15,22,23,30,24,37,34,44)(16,26,28,42,27,41,29,43)(21,35,25,38,36,40,39,45);(16,27)(21,36)(25,39)(26,41)(28,29)(35,40)(38,45)(42,43)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13756);