Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25927 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,10,13)(8,12,9,11)(14,15,18,23)(16,25,27,21)(17,29,31,41)(19,33,20,24)(22,35,40,32)(26,43,44,39)(28,37,34,38)(30,36,45,42);(6,8,10,9)(7,11,13,12)(14,16,18,27)(15,21,23,25)(17,30,31,45)(19,28,20,34)(22,39,40,43)(24,37,33,38)(26,35,44,32)(29,42,41,36);(6,8,10,9)(7,12,13,11)(14,17,20,35,18,31,19,32)(15,22,33,41,23,40,24,29)(16,26,34,30,27,44,28,45)(21,36,38,43,25,42,37,39);(6,9,10,8)(7,12,13,11)(15,23)(17,31)(21,25)(24,33)(26,44)(30,45)(32,35)(37,38)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18915);