Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1972 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
414 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)(8,10,16,28)(9,21)(11,30,29,15)(13,20,27,18)(14,31)(19,26)(22,25)(23,32)(24,33)(34,35)(36,37);(6,8,11,18,13,15,10,17,12,16,29,20,27,30,28,7)(9,22,31,14,25,21,35,32,26,36,24,33,37,19,23,34);(6,9,11,24,13,25,10,23,12,26,29,31,27,37,28,35)(7,14,16,21,18,32,30,36,17,33,8,19,20,34,15,22)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6235);