Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33427 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#432 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C5×#405 ◁ G | |
524 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)(6,7,9,14,10,15,22,30,11,12,20,25,23,26,35,37)(8,16,18,32,19,24,34,29,17,31,21,36,33,13,27,28);(7,12)(9,20)(13,24)(15,26)(16,31)(18,21)(22,35)(27,34);(6,8)(7,13,12,24)(9,21)(10,19)(11,17)(14,29,25,28)(15,16,26,31)(18,20)(22,27)(23,33)(30,36,37,32)(34,35);(8,17)(13,24)(16,31)(18,21)(19,33)(27,34)(28,29)(32,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26415);