Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#9587 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #388 |
| prime radical | C23×C5 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,11)(8,14)(9,17)(12,20)(13,19)(15,18)(22,23,26,30,27,31,25,28)(24,32,35,29,33,36,34,37);(6,7,9,11,10,13,17,19)(8,15,16,20,14,12,21,18)(23,28,31,30)(24,33)(29,36,37,32)(34,35);(6,8,9,16,10,14,17,21)(7,12,11,18,13,15,19,20)(22,24,25,34,27,33,26,35)(23,29,30,36,31,37,28,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2575);