Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#21733 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C23×C5 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
654 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)(8,13)(9,11)(10,12)(14,15,20,25,21,26,19,36)(16,22,30,38,31,39,29,44)(17,32,18,24,34,41,33,23)(27,42,35,40,43,45,28,37);(6,8,10,9)(7,11,12,13)(14,16)(15,22)(17,27)(18,35)(19,29)(20,30)(21,31)(23,37)(24,40)(25,38)(26,39)(28,33)(32,42)(34,43)(36,44)(41,45);(14,17,19,33,21,34,20,18)(15,23,25,32,26,24,36,41)(16,27,29,28,31,43,30,35)(22,37,38,42,39,40,44,45);(6,9,10,8)(7,11,12,13)(14,18,20,34,21,33,19,17)(15,24,25,41,26,23,36,32)(16,28,30,27,31,35,29,43)(22,37,38,42,39,40,44,45)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14721);