Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#8877 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C5×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,17)(8,18,9,20)(10,16,11,15)(13,21,14,19)(22,23,26,31,28,25,27,30)(24,32,35,33,36,37,34,29);(6,8,12,9)(7,13,17,14)(10,21,11,19)(15,20,16,18)(22,24,26,35,28,36,27,34)(23,29,31,32,25,33,30,37);(6,9,12,8)(7,14,17,13)(10,21,11,19)(15,20,16,18)(22,25,28,23)(24,33,36,29)(26,30,27,31)(32,35,37,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1865);