Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9263 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
478 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)(7,12,14,21)(8,16,18,35)(9,20)(15,27,26,36)(17,34,22,19)(24,33)(25,32)(28,29);(6,7,9,21,23,26,33,27)(8,13,22,28,34,37,16,32)(10,15,20,36,11,14,24,12)(17,25,19,30,35,29,18,31);(6,8)(7,13)(9,22)(10,19)(11,18)(12,25)(14,31)(15,30)(16,33)(17,24)(20,35)(21,28)(23,34)(26,37)(27,32)(29,36)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2251);