Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C5×#62823 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
6538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 37 points:
(1,2,3,4,5)(6,7,12,13)(8,16,17,14)(9,19,18,15)(10,11,20,21)(23,27)(25,32)(28,33)(31,36);(7,13)(8,17)(10,20)(15,19)(24,30)(28,33)(31,36)(35,37);(6,8,12,17)(7,14,13,16)(9,10,18,20)(11,19,21,15)(22,23,26,27)(24,28,30,33)(25,29,32,34)(31,35,36,37);(6,9,12,18)(7,15,13,19)(8,10,17,20)(11,16,21,14);(6,10)(7,11)(8,18)(9,17)(12,20)(13,21)(14,19)(15,16)(22,24)(23,28)(25,31)(26,30)(27,33)(29,35)(32,36)(34,37);(6,11)(7,10)(8,19)(9,16)(12,21)(13,20)(14,18)(15,17)(22,25)(23,29)(24,31)(26,32)(27,34)(28,35)(30,36)(33,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55811);