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×#60327 |
| 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 | |
1812 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)(14,15,18,22,19,23,30,37)(16,20,26,33,24,31,38,42)(17,21,29,36,28,35,41,45)(25,32,39,43,27,34,40,44);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,27)(18,26)(19,24)(21,34)(22,33)(23,31)(25,28)(29,40)(30,38)(32,35)(36,44)(37,42)(39,41)(43,45);(8,13)(10,12)(16,24)(17,28)(20,31)(21,35)(26,38)(29,41)(33,42)(36,45);(7,11)(10,12)(16,24)(20,31)(25,27)(26,38)(32,34)(33,42)(39,40)(43,44);(14,17,18,29,19,28,30,41)(15,21,22,36,23,35,37,45)(16,25,26,39,24,27,38,40)(20,32,33,43,31,34,42,44)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53315);