Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13259 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1982 |
| prime radical | C5×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
472 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)(6,7,11,18,12,17,27,36,13,19,26,15,10,28,9,16)(8,20,24,34,23,35,22,31,25,30,21,14,29,32,37,33);(6,8,12,29,13,25,10,23)(7,14,17,31,19,34,28,33)(9,21,11,22,27,24,26,37)(15,30,16,35,18,20,36,32);(6,9,10,26,13,27,12,11)(7,15,17,16,19,18,28,36)(8,21,23,37,25,24,29,22)(14,30,31,35,34,20,33,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6247);