Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
656 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)(7,12)(8,16)(9,21)(10,22)(14,29)(15,26)(19,20)(23,34)(24,28)(27,35)(30,32)(31,37);(6,7,10,12,11,15,22,26)(8,17,16,24,20,13,19,28)(9,14,21,25,23,29,34,36)(18,31,30,35,33,27,32,37);(6,8)(7,13)(9,18)(10,19)(11,20)(12,24)(14,27)(15,17)(16,22)(21,32)(23,33)(25,35)(26,28)(29,31)(30,34)(36,37);(6,9,22,34,11,23,10,21)(7,14,26,36,15,29,12,25)(8,18,16,30,20,33,19,32)(13,27,28,37,17,31,24,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17976);