Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11092 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #394 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,21,16,20,14,19)(12,17,25,33,24,34,23,32)(13,26,30,18,28,36,29,35)(22,37,39,31,27,40,38,41);(10,12)(11,17)(13,27,28,22)(14,23)(15,25)(16,24)(18,31,35,41)(19,32)(20,34)(21,33)(26,37,36,40)(29,39,30,38);(7,9)(10,13)(11,18)(12,22)(14,30)(15,29)(16,28)(17,31)(19,36)(20,35)(21,26)(23,39)(24,27)(25,38)(32,40)(33,37)(34,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4080);