Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C5×#3010 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,25)(17,28)(18,21)(19,23)(20,34)(22,31)(24,33)(26,42)(27,37)(29,45)(30,39)(32,41)(35,38)(36,44)(40,43);(7,10)(11,13)(14,16,19,18)(15,21,23,25)(17,26,30,29)(20,27,33,32)(22,36,38,40)(24,37,34,41)(28,45,39,42)(31,43,35,44);(14,17,19,30)(15,22,23,38)(16,26,18,29)(20,35,33,31)(21,36,25,40)(24,39,34,28)(27,44,32,43)(37,42,41,45);(6,8,9,12)(7,11,10,13)(14,18,19,16)(15,21,23,25)(17,29,30,26)(20,32,33,27)(22,36,38,40)(24,37,34,41)(28,45,39,42)(31,44,35,43);(16,18)(21,25)(26,29)(27,32)(36,40)(37,41)(42,45)(43,44)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53583);