Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33329 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×C42 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C22×#97 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
716 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)(6,7,9,14,10,16,21,29,11,15,20,30,22,12,23,25)(8,13,18,27,19,28,33,37,17,26,31,36,32,24,34,35);(7,12,15,16)(9,20)(13,24,26,28)(14,29,30,25)(18,31)(21,23)(27,37,36,35)(33,34);(6,8)(7,13)(9,18)(10,19)(11,17)(12,24)(14,27)(15,26)(16,28)(20,31)(21,33)(22,32)(23,34)(25,35)(29,37)(30,36);(8,17)(13,26)(18,31)(19,32)(24,28)(27,36)(33,34)(35,37)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26317);