Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,9,13)(8,16,18,12,19,20,17,21)(22,23,26,30,27,29,36,24)(25,32,34,37,33,28,35,31);(6,8)(7,12)(9,18)(10,17)(11,19)(13,20)(14,16)(15,21)(22,24,26,29,27,30,36,23)(25,31,35,32,33,37,34,28);(6,8)(7,12)(9,17)(10,18)(11,19)(13,16)(14,20)(15,21)(22,25)(23,28)(24,31)(26,35)(27,33)(29,32)(30,37)(34,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2325);