Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12951 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1864 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
406 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)(6,7,11,18,13,19,12,31)(8,20,23,32,25,36,24,35)(9,26,10,29,28,17,27,16)(14,22,33,21,34,37,15,30);(6,8)(7,14,19,34)(9,22)(10,30)(11,24)(12,23)(13,25)(15,31,33,18)(16,32,29,35)(17,36,26,20)(21,27)(28,37);(6,8)(7,15)(9,22)(10,30)(11,24)(12,23)(13,25)(14,31)(16,20)(17,32)(18,34)(19,33)(21,27)(26,35)(28,37)(29,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5939);