Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#29507 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
758 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)(8,12)(9,11)(10,13)(24,30)(25,31)(27,33)(29,34)(32,36)(35,37);(6,8,9,13)(7,10,11,12)(14,15)(16,19)(17,18)(20,21)(22,23,26,28)(24,27,32,35)(25,29,31,34)(30,33,36,37);(15,18)(19,21)(22,24,31,36,26,32,25,30)(23,27,34,37,28,35,29,33);(7,11)(10,12)(14,16,17,20)(15,19,18,21)(23,28)(27,35)(29,34)(33,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22495);