Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C5×#7462 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | D16 |
| commutator | C16 | C2×C5×C8 |
| Frattini, cosocle | C4×C16 | C22×C5 |
| socle | C2×C5 | #1371 |
| prime radical | C22×C5 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×C8 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
78 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)(6,7,10,16,12,17,31,43,13,19,32,44,34,18,11,33)(8,20,23,46,25,47,55,67,26,49,56,68,58,48,24,57)(9,27,28,40,29,59,52,42,30,61,62,41,51,60,21,15)(14,35,36,53,37,63,65,54,39,64,66,69,38,50,45,22);(6,8,21,50,11,24,51,69,34,58,62,64,32,56,30,54,13,26,52,63,31,55,29,53,12,25,28,35,10,23,9,22)(7,14,27,46,16,36,40,47,17,37,59,67,43,65,42,49,19,39,61,68,44,66,41,48,18,38,60,57,33,45,15,20)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 450);