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×#11086 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #397 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
866 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)(10,11,15,21,16,19,14,20)(12,17,25,33,23,34,24,32)(13,26,30,36,28,18,29,37)(22,31,39,40,38,41,27,35);(6,7)(8,9)(10,12)(11,17)(13,27,28,39)(14,24)(15,25)(16,23)(18,35,26,40)(19,34)(20,32)(21,33)(22,29,38,30)(31,36,41,37);(7,9)(10,13)(11,18)(12,22)(14,30)(15,29)(16,28)(17,31)(19,26)(20,37)(21,36)(23,38)(24,39)(25,27)(32,40)(33,35)(34,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4074);