Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#392 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
960 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)(9,19)(13,25)(14,23)(15,20)(18,30)(21,32)(24,27)(26,35)(28,33)(34,36);(6,7,10,15)(8,13,16,27)(9,20,19,12)(11,14,22,28)(17,25,29,24)(18,26,30,36)(21,33,32,23)(31,35,37,34);(9,21)(12,23)(17,31)(19,32)(20,33)(24,34)(25,35)(29,37);(6,8,11,18)(7,13,14,26)(9,17,21,31)(10,16,22,30)(12,24,23,34)(15,27,28,36)(19,29,32,37)(20,25,33,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11142);