Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C16 · Schur multiplier (H2) = C25 |
| direct product | C5×#33496 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C2 | C23×C5×C16 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C23×C5 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,24,27,25,28,30,33,26,29,31,34,32,35,36,37);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20);(7,13)(11,14)(12,21)(19,20)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26484);