Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8504 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C5×#3529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,10,11,12)(14,15)(16,24,27,20)(17,28,29,39)(18,23)(19,22)(21,30,36,32)(25,41)(26,38,31,35)(33,37)(34,43)(40,44)(42,45);(6,7)(8,12)(9,11)(10,13)(14,16)(15,20)(17,25)(18,31)(19,27)(21,34)(22,24)(23,38)(26,33)(28,41)(29,42)(30,44)(32,43)(35,37)(36,40)(39,45);(8,13)(10,12)(14,17)(15,21)(16,25)(18,32)(19,29)(20,34)(22,36)(23,39)(24,40)(26,44)(27,42)(28,37)(30,33)(31,43)(35,41)(38,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1492);