Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C5×#7464 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #367 |
| commutator | C16 | C2×C5×C8 |
| Frattini, cosocle | C4×C16 | C22×C5 |
| socle | C2×C5 | #1372 |
| prime radical | C22×C5 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7,10,16,12,17,33,43,13,19,28,47,35,18,11,34)(8,20,23,49,25,51,24,61,26,53,32,50,62,52,63,69)(9,27,21,44,30,64,54,46,31,65,55,45,60,37,29,15)(14,36,38,67,40,68,66,22,42,56,39,58,41,57,48,59);(6,8,21,36,11,24,9,22,35,62,29,57,28,23,60,67,13,26,55,56,33,63,31,59,12,25,54,68,10,32,30,58)(7,14,37,61,16,39,15,20,17,40,27,50,43,48,44,51,19,42,64,69,47,38,46,53,18,41,65,49,34,66,45,52)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 452);