Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#22318 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15)(16,19)(17,18)(20,21)(22,23,26,30,27,24,28,32)(25,29,34,36,33,31,35,37);(6,8)(7,10)(9,13)(11,12)(22,24,26,32,27,23,28,30)(25,31,34,37,33,29,35,36);(8,13)(10,12)(15,18)(19,21)(22,25)(23,29)(24,31)(26,34)(27,33)(28,35)(30,36)(32,37);(7,11)(10,12)(14,16,17,20)(15,19,18,21)(25,33)(29,31)(34,35)(36,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15306);