Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#9370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3219 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
424 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)(8,14)(9,17,19,12)(10,16)(11,13)(15,18,21,20)(22,23,26,29)(24,27)(25,30,34,35)(28,33)(31,37)(32,36);(6,8,14,7,11,16,10,13)(9,15,18,12,19,21,20,17)(22,24)(23,27,29,37)(25,33)(26,31)(28,35,36,30)(32,34);(6,9)(7,12)(8,15)(10,20)(11,19)(13,17)(14,18)(16,21)(22,24)(23,28)(25,33)(26,31)(27,35)(29,36)(30,37)(32,34)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2358);