Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C4 |
| direct product | C5×#13106 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,20,25,19,26,36,43,18,23,35,27,34,24,41,42)(16,17,30,32,29,33,40,45,28,21,37,38,44,31,22,39);(7,10)(11,13)(14,16,20,30,19,29,36,40,18,28,35,37,34,44,41,22)(15,21,25,38,26,31,43,39,23,17,27,32,24,33,42,45);(6,8,9,12)(7,11,10,13)(14,17,18,21)(15,22,23,40)(16,27,28,25)(19,33,34,31)(20,32,35,38)(24,37,26,30)(29,42,44,43)(36,45,41,39)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6094);