Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17602 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3522 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14,16,21)(9,13)(10,12)(11,15,19,17)(18,20)(22,23)(24,30)(25,34)(26,31)(27,36)(28,32)(29,33)(35,37);(6,8)(7,11)(9,17)(10,16)(12,19)(13,21)(14,20)(15,18)(22,24,28,25)(23,29,32,37)(26,33,27,35)(30,31,34,36);(7,12)(11,19)(13,20)(14,21)(22,25,28,24)(23,30,32,34)(26,33,27,35)(29,31,37,36);(22,26,28,27)(23,31,32,36)(24,33,25,35)(29,34,37,30)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10590);