Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
538 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)(6,7,10,17,11,18,20,30)(8,13,12,19,22,15,27,34)(9,23,26,29,25,35,36,33)(14,31,32,37,16,21,28,24);(7,13,18,15)(9,24)(14,28,16,32)(17,19,30,34)(21,36)(23,33,35,29)(25,37)(26,31);(7,14)(8,20)(9,25)(10,22)(13,28)(15,32)(16,18)(17,29)(19,23)(21,26)(24,37)(30,33)(31,36)(34,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2513);