Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#9586 |
| 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 | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
312 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)(8,15)(9,12)(13,20)(14,18)(17,21)(22,23,25,28,27,31,26,30)(24,32,33,37,35,36,34,29);(6,7,12,18,10,14,9,11)(8,13,19,20,15,21,16,17)(23,28,31,30)(29,32,37,36);(6,8,9,16,10,15,12,19)(7,13,11,17,14,21,18,20)(22,24,25,33,27,35,26,34)(23,29,30,36,31,37,28,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2574);