Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60300 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#3475 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,13)(10,12)(11,15)(16,20)(17,19)(18,21)(22,23,27,31,26,28,25,30)(24,32,33,37,34,29,35,36);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(23,28)(29,32)(30,31)(36,37);(8,15)(11,14)(16,21)(18,20)(23,28)(29,32)(30,31)(36,37);(7,12)(11,14)(13,19)(18,20)(22,24)(23,29)(25,35)(26,34)(27,33)(28,32)(30,37)(31,36);(6,9,10,17)(7,13,12,19)(8,16,15,21)(11,18,14,20)(22,25,26,27)(23,30,28,31)(24,33,34,35)(29,36,32,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53288);