Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1936 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,13,17,24)(8,19,22,27)(9,14)(10,25)(11,28)(15,30)(16,33,26,18)(21,35,31,23)(29,34)(32,36);(6,7,14,16,28,33,9,24,12,17,10,26,11,18,25,13)(8,20,27,36,35,15,31,32,22,37,19,29,23,30,21,34);(6,8,9,21,11,23,14,19,12,22,25,31,28,35,10,27)(7,15,13,29,18,20,26,32,17,30,24,36,33,37,16,34)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6144);