Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13027 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1879 |
| prime radical | C5×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
484 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)(6,7,11,16,12,18,29,36,13,19,9,26,28,31,10,17)(8,20,23,33,24,32,22,34,25,37,21,15,30,35,27,14);(6,8)(7,14,19,34)(9,27)(10,21)(11,22)(12,30)(13,25)(15,31,33,18)(16,32,26,35)(17,37,36,20)(23,29)(24,28);(6,8)(7,15)(9,27)(10,21)(11,22)(12,30)(13,25)(14,31)(16,20)(17,32)(18,34)(19,33)(23,29)(24,28)(26,37)(35,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6015);