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×#13158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1936 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
316 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)(6,7,12,17,14,18,13,28)(8,10,16,29,21,19,31,11)(9,22,26,37,27,23,32,20)(15,33,34,25,36,24,35,30);(6,8,19,17,13,16,29,18,14,21,10,28,12,31,11,7)(9,23,24,15,26,37,30,35,27,22,33,36,32,20,25,34);(6,9,10,24,13,26,11,30,14,27,19,33,12,32,29,25)(7,15,8,20,17,35,16,23,18,36,21,37,28,34,31,22)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6146);