Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11000 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1438 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,20,22,13)(10,23)(11,15)(12,18)(14,25)(17,31,33,37)(19,30)(21,29,36,34)(24,27)(26,35,28,32);(6,8)(7,12)(9,17)(10,24)(11,19)(13,28)(14,30)(15,27)(16,23)(18,25)(20,26)(21,35)(22,33)(29,31)(32,36)(34,37);(6,9)(7,13)(8,17)(10,21)(11,22)(12,26)(14,29)(15,20)(16,31)(18,32)(19,33)(23,34)(24,35)(25,36)(27,28)(30,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3988);