Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#22951 |
| 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 | C5×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×C22⋊C8 ◁ G | |
730 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)(7,10)(11,13)(14,15,19,23,20,24,18,32)(16,21,28,36,26,35,27,42)(17,29,31,22,30,38,33,37)(25,39,40,34,41,44,43,45);(6,7)(8,11)(9,10)(12,13)(14,16,19,28,20,26,18,27)(15,21,23,36,24,35,32,42)(17,25,31,40,30,41,33,43)(22,34,38,44,37,45,29,39);(6,8,9,12)(7,11,10,13)(14,17)(15,22)(16,25)(18,33)(19,31)(20,30)(21,34)(23,38)(24,37)(26,41)(27,43)(28,40)(29,32)(35,45)(36,44)(39,42);(7,10)(11,13)(16,26)(17,30)(21,35)(22,37)(27,28)(29,38)(31,33)(36,42)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15939);