Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,15)(8,16,18,21)(9,14,10,13)(12,19,20,17)(22,23,27,31)(24,32,35,36)(25,30,26,29)(28,34,37,33);(6,8,11,18)(7,12,15,20)(9,19,10,17)(13,21,14,16)(22,24,27,35)(23,28,31,37)(25,34,26,33)(29,36,30,32);(6,8)(7,12)(9,17)(10,19)(11,18)(13,16)(14,21)(15,20)(24,33)(25,26)(28,32)(29,30)(34,35)(36,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3186);