Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22286 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2328 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
820 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)(8,11)(9,20,24,18)(10,25)(12,16)(13,17)(14,23,30,22)(15,26)(19,28)(21,34,37,29)(27,31)(32,35,33,36);(6,8,12,17,13,16,11,7)(9,14,22,20,24,30,23,18)(10,19,27,31,28,25,26,15)(21,29,35,33,37,34,36,32);(6,9)(7,14,17,30)(8,18,16,20)(10,21)(11,22)(12,23)(13,24)(15,29,31,34)(19,32,25,33)(26,35)(27,36)(28,37);(6,10,12,27,13,28,11,26)(7,15,8,19,17,31,16,25)(9,21,23,36,24,37,22,35)(14,29,18,32,30,34,20,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15274);