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×#33426 |
| 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 | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#97 ◁ G | |
520 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)(6,7,9,15,10,16,23,32,11,14,22,30,24,13,28,29)(8,17,19,31,20,25,34,36,18,12,21,26,33,27,35,37);(6,8)(7,12)(9,21)(10,20)(11,18)(13,25)(14,17)(15,31)(16,27)(19,22)(23,35)(24,33)(26,30)(28,34)(29,37)(32,36);(7,13,14,16)(9,22)(12,25,17,27)(15,32,30,29)(19,21)(23,28)(26,37,31,36)(34,35);(8,18)(12,17)(19,21)(20,33)(25,27)(26,31)(34,35)(36,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26414);