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×#60598 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
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)(16,21)(17,23)(18,24)(19,33)(20,25)(22,29)(26,35)(27,36)(28,38)(30,39)(31,37)(32,40)(34,42)(41,44)(43,45);(7,10)(11,13)(14,16,20,28)(15,21,25,38)(17,26,30,43)(18,31,19,27)(22,34,40,44)(23,35,39,45)(24,37,33,36)(29,42,32,41);(14,17)(15,22)(16,26)(18,29)(19,32)(20,30)(21,34)(23,33)(24,39)(25,40)(27,41)(28,43)(31,42)(35,36)(37,45)(38,44);(6,8,9,12)(7,11,10,13)(14,17)(15,23)(16,26)(18,32)(19,29)(20,30)(21,35)(22,33)(24,40)(25,39)(27,42)(28,43)(31,41)(34,36)(37,44)(38,45);(14,18,20,19)(15,24,25,33)(16,27,28,31)(17,29,30,32)(21,36,38,37)(22,39,40,23)(26,41,43,42)(34,35,44,45)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53586);