Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#9240 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
452 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)(8,11,15,18)(9,12)(10,13)(14,17,20,21)(16,19)(23,27,29,32)(24,30,31,33);(6,8)(7,11,13,18)(9,14)(10,15)(12,17,19,21)(16,20)(22,23,25,32,26,29,34,27)(24,28,33,36,31,37,30,35);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,17,18,21)(22,24)(23,28)(25,33)(26,31)(27,35)(29,37)(30,34)(32,36)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2228);