Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13129 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1924 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ 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)(6,7,11,16,12,17,28,19,10,15,18,22,25,31,33,35)(8,9,14,23,20,24,30,34,21,13,26,27,32,29,36,37);(6,8,18,14,12,20,33,30,10,21,11,26,25,32,28,36)(7,13,16,23,17,29,19,34,15,9,22,27,31,24,35,37);(6,9,10,13)(7,14)(8,19)(11,27,18,23)(12,24,25,29)(15,26)(16,32)(17,30)(20,22)(21,35)(28,37,33,34)(31,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6117);