Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8321 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C5×#537 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
710 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)(7,11)(9,16)(12,20)(13,17)(14,18)(19,21)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(6,7)(8,12)(9,17)(10,13)(11,16)(14,19)(15,21)(18,20)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(6,8,10,15)(7,12,13,21)(9,18,16,14)(11,19,17,20)(24,30)(27,33)(31,36)(34,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1309);