Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#538 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
1080 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)(6,7,9,14,11,15,20,28)(8,13,18,26,16,27,30,37)(10,12,19,23,21,24,33,35)(17,22,31,34,29,25,32,36);(7,12)(8,16)(9,19)(13,25)(15,24)(17,29)(18,32)(20,33)(22,27)(26,37)(30,31)(34,36);(6,8)(7,13)(9,18)(10,17)(11,16)(12,22)(14,26)(15,27)(19,31)(20,30)(21,29)(23,34)(24,25)(28,37)(32,33)(35,36);(8,17)(13,22)(16,29)(18,31)(25,27)(26,34)(30,32)(36,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10374);