Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C25×C5 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C42×C5 ◁ G | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,15,11,14,21,28)(8,13,18,27,16,26,29,37)(10,12,20,24,17,23,31,35)(19,22,33,34,30,25,32,36);(7,12)(8,16)(9,20)(13,25)(14,23)(18,32)(19,30)(21,31)(22,26)(27,37)(29,33)(34,36);(6,8,17,30)(7,13,23,25)(9,18,31,32)(10,19,11,16)(12,22,14,26)(15,27,35,36)(20,33,21,29)(24,34,28,37);(7,14)(12,23)(13,26)(15,28)(22,25)(24,35)(27,37)(34,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10386);