Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#60612 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C23×D4 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
2050 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)(16,26)(17,31)(18,23)(19,24)(20,25)(21,29)(22,33)(27,35)(28,37)(30,36)(32,39)(34,38)(40,43)(41,45)(42,44);(7,10)(11,13)(14,16,19,30,20,28,18,29)(15,21,23,37,25,36,24,26)(17,27,33,43,34,41,32,42)(22,35,31,44,39,45,38,40);(6,8,9,12)(7,11,10,13)(14,17,18,32,20,34,19,33)(15,22,24,38,25,39,23,31)(16,27,29,42,28,41,30,43)(21,35,26,40,36,45,37,44);(16,28)(21,36)(26,37)(27,41)(29,30)(35,45)(40,44)(42,43);(14,18,20,19)(15,23,25,24)(16,29,28,30)(17,32,34,33)(21,37,36,26)(22,31,39,38)(27,42,41,43)(35,44,45,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53600);