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×#62839 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
6616 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)(7,11)(12,17)(13,18)(19,21)(22,23)(24,33)(25,26)(27,35)(28,29)(30,34)(31,37)(32,36);(9,16)(13,18)(14,20)(19,21)(23,29)(24,34)(26,31)(32,35);(6,7)(8,12)(9,13)(10,11)(14,19)(15,17)(16,18)(20,21)(22,24)(23,30)(25,35)(26,36)(27,31)(28,34)(29,33)(32,37);(6,8,10,15)(7,12,11,17)(9,14,16,20)(13,19,18,21)(22,25)(23,31)(24,35)(26,29)(27,30)(28,37)(32,34)(33,36);(6,9,10,16)(7,13,11,18)(8,14,15,20)(12,19,17,21)(22,26,28,31)(23,25,29,37)(24,36,34,27)(30,35,33,32);(6,7)(8,12)(9,13)(10,11)(14,19)(15,17)(16,18)(20,21)(22,27,28,36)(23,32,29,35)(24,31,34,26)(25,30,37,33)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55827);