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) = C27 |
| direct product | C5×#20360 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C22×#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×#41 ◁ G | |
1996 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)(8,16,18,12)(9,13)(10,15)(11,14)(17,28,30,23)(19,29,32,24)(20,25)(21,26)(22,27)(31,36,37,34)(33,35);(6,8,10,19,11,18,22,32)(7,12,15,24,14,16,27,29)(9,17,21,31,20,30,33,37)(13,23,26,34,25,28,35,36);(6,9)(7,13)(8,17)(10,21)(11,20)(12,23)(14,25)(15,26)(16,28)(18,30)(19,31)(22,33)(24,34)(27,35)(29,36)(32,37);(9,20)(13,25)(17,30)(21,33)(23,28)(26,35)(31,37)(34,36)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13348);