Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#20401 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×C4×C22⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×M16 ◁ G | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,22,28)(8,16,19,30,18,12,23,25)(9,13,21,27,20,26,33,36)(17,29,32,37,31,24,34,35);(6,8)(7,12,15,16)(9,17)(10,19)(11,18)(13,24,26,29)(14,25,28,30)(20,31)(21,32)(22,23)(27,35,36,37)(33,34);(6,9,10,21,11,20,22,33)(7,13,14,27,15,26,28,36)(8,17,19,32,18,31,23,34)(12,24,25,35,16,29,30,37);(9,20)(13,26)(17,31)(21,33)(24,29)(27,36)(32,34)(35,37)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13389);