Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12612 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(14,15,17,21,18,16,23,24,19,22,26,29,27,25,20,28);(7,9)(10,11,12,13);(14,16,17,24,18,22,23,29,19,25,26,28,27,15,20,21)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5600);