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×#60318 |
| 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)(14,15,18,23,19,24,31,39)(16,20,27,33,25,21,34,36)(17,28,29,42,30,22,37,38)(26,35,41,45,40,32,43,44);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,26)(18,27)(19,25)(21,24)(22,32)(23,33)(28,35)(29,41)(30,40)(31,34)(36,39)(37,43)(38,44)(42,45);(8,13)(10,12)(14,16)(15,21)(17,26)(18,27)(19,25)(20,24)(22,35)(23,36)(28,32)(29,41)(30,40)(31,34)(33,39)(37,43)(38,45)(42,44);(7,11)(10,12)(16,25)(20,21)(26,40)(27,34)(32,35)(33,36)(41,43)(44,45);(14,17,18,29,19,30,31,37)(15,22,23,38,24,28,39,42)(16,26,27,41,25,40,34,43)(20,32,33,44,21,35,36,45)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53306);