Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1804 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
390 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)(6,7,9,15,12,17,11,29)(8,18,21,13,22,32,19,33)(10,24,23,30,28,37,27,16)(14,26,35,25,31,36,34,20);(6,8)(7,13,17,33)(9,19)(10,25)(11,21)(12,22)(14,30,31,16)(15,18,29,32)(20,28)(23,26)(24,34,37,35)(27,36);(6,8)(7,14)(9,21)(10,26)(11,19)(12,22)(13,30)(15,35)(16,33)(17,31)(18,24)(20,27)(23,25)(28,36)(29,34)(32,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5816);