Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60315 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1816 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)(8,12)(9,11)(10,13)(14,15,18,23,19,21,30,34)(16,20,26,33,24,31,38,42)(17,27,28,41,29,22,35,36)(25,39,40,45,37,32,43,44);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,25)(18,26)(19,24)(21,31)(22,32)(23,33)(27,39)(28,40)(29,37)(30,38)(34,42)(35,43)(36,44)(41,45);(8,13)(10,12)(15,21)(16,24)(22,27)(23,34)(25,37)(26,38)(36,41)(40,43);(7,11)(10,12)(16,24)(20,31)(25,37)(26,38)(32,39)(33,42)(40,43)(44,45);(14,17,18,28,19,29,30,35)(15,22,23,36,21,27,34,41)(16,25,26,40,24,37,38,43)(20,32,33,44,31,39,42,45)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53303);