Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13127 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #388 |
| 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 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
300 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)(6,7,11,17,10,15,19,14,12,16,28,34,26,35,9,24)(8,18,20,33,22,13,25,31,23,32,27,36,21,30,37,29);(6,8,19,37,26,21,11,27,12,23,9,25,10,22,28,20)(7,13,14,33,35,18,17,29,16,30,24,36,15,32,34,31);(6,9,10,11,12,19,26,28)(7,14,15,34,16,24,35,17)(8,20,21,37,23,27,22,25)(13,29,18,36,30,31,32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6115);