Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11956 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #333 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C2×C5 | C2×#333 |
| prime radical | C23×C5 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
292 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)(6,7,10,14,23,15,33,31,11,16,25,32,9,13,22,29)(8,17,20,30,18,28,24,36,19,34,35,37,21,12,26,27);(6,8,11,19)(7,12)(9,18,23,21)(10,24,25,26)(13,17)(14,30)(15,34)(16,28)(20,22,35,33)(27,31)(29,36)(32,37);(6,9,11,23)(7,13,16,15)(8,18,19,21)(10,22,25,33)(12,17,28,34)(14,29,32,31)(20,24,35,26)(27,30,36,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4944);