Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#7454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1365 |
| commutator | C2×C16 | C2×C4×C5 |
| Frattini, cosocle | C4×C16 | C22×C5 |
| prime radical | C22×C5 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
96 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)(6,7,10,17,35,19,30,16,11,38,36,48,41,47,33,67,14,20,31,52,12,39,32,49,13,18,37,51,29,54,9,28)(8,21,23,43,62,46,61,55,24,50,63,15,40,42,22,44,27,58,65,53,25,66,60,56,26,57,64,45,59,69,34,68);(6,8)(7,15,20,45)(9,23)(10,34)(11,26)(12,40)(13,24)(14,27)(16,46,49,66)(17,50,52,57)(18,53,38,43)(19,55,39,56)(21,51,58,48)(22,31)(25,41)(28,42,67,69)(29,62)(30,64)(32,63)(33,65)(35,59)(36,60)(37,61)(44,47,68,54)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 442);