Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#22456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,21,23,13)(10,24)(11,15)(12,19)(14,25)(17,31,34,30)(18,28)(20,27)(22,29,37,36)(26,33,35,32);(6,8,11,20)(7,12,15,28)(9,17,23,34)(10,18,25,19)(13,26,21,35)(14,27,24,16)(22,32,37,33)(29,31,36,30);(6,9)(7,13)(8,17)(10,22)(11,23)(12,26)(14,29)(15,21)(16,30)(18,32)(19,33)(20,34)(24,36)(25,37)(27,31)(28,35);(8,18)(10,25)(12,27)(14,24)(16,28)(17,32)(19,20)(22,37)(26,31)(29,36)(30,35)(33,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15444);