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 |
| direct product | C5×#13182 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1947 |
| prime radical | C23×C5 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
280 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)(6,7,12,17,13,18,11,28)(8,19,22,35,23,29,21,33)(9,16,10,26,25,34,24,15)(14,27,30,37,32,36,31,20);(6,8,12,22,13,23,11,21)(7,14,17,30,18,32,28,31)(9,20,10,27,25,37,24,36)(15,29,16,33,26,19,34,35);(6,9,11,24,13,25,12,10)(7,15,17,16,18,26,28,34)(8,20,21,36,23,37,22,27)(14,29,30,33,32,19,31,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6170);