Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12634 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | C4×#99 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
452 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)(6,7,11,9,12,15,26,22)(8,16,19,30,20,31,34,37)(10,23,24,35,25,14,29,21)(13,27,17,32,28,36,33,18);(6,8)(7,13)(9,17)(10,18,25,32)(11,19)(12,20)(14,16,23,31)(15,28)(21,30,35,37)(22,33)(24,27,29,36)(26,34);(6,9,11,15,12,22,26,7)(8,17,19,28,20,33,34,13)(10,21,24,23,25,35,29,14)(16,18,30,27,31,32,37,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5622);