Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18298 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1024 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)(7,14)(8,19)(9,24)(10,20)(15,33)(16,35)(17,26)(18,32)(21,25)(22,23)(27,28)(29,37)(30,31)(34,36);(6,7)(8,15)(9,16)(10,26)(11,29)(12,17)(13,18)(14,20)(19,35)(21,36)(22,31)(23,34)(24,30)(25,33)(27,32)(28,37);(6,8,13,23)(7,15,18,34)(9,20,24,27)(10,25,28,21)(11,19,12,22)(14,30,32,16)(17,31,29,35)(26,33,37,36);(6,9,12,21)(7,16,17,36)(8,20,22,10)(11,25,13,24)(14,31,26,15)(18,30,29,33)(19,28,23,27)(32,35,37,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11286);