Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11393 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
850 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)(14,15,19,23,20,24,18,31)(16,25,27,21,28,36,32,35)(17,22,29,38,30,39,33,37)(26,41,44,45,43,40,42,34);(6,7,9,11)(8,12,13,10)(14,16)(15,21)(17,26)(18,32)(19,27)(20,28)(22,34)(23,36)(24,35)(25,31)(29,44)(30,43)(33,42)(37,40)(38,41)(39,45);(6,8,9,13)(7,10,11,12)(14,17)(15,22)(16,26)(18,33)(19,29)(20,30)(21,34)(23,38)(24,39)(25,40)(27,44)(28,43)(31,37)(32,42)(35,45)(36,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4381);