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) = C23 |
| direct product | C5×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1802 |
| 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 | |
400 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)(6,7,12,16,13,18,10,26)(8,17,11,27,21,34,20,14)(9,22,24,37,25,29,23,33)(15,28,30,36,32,35,31,19);(6,8,10,20,13,21,12,11)(7,14,16,17,18,27,26,34)(9,19,23,35,25,36,24,28)(15,29,30,33,32,22,31,37);(6,9)(7,15)(8,19)(10,23)(11,28)(12,24)(13,25)(14,29)(16,30)(17,33)(18,32)(20,35)(21,36)(22,27)(26,31)(34,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5810);