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) = C24 |
| direct product | C5×#11350 |
| 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 | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
594 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)(12,20)(13,24)(14,27)(15,19)(16,22)(17,26)(18,29)(21,37)(23,36)(25,33)(28,34)(30,38)(31,41)(32,40)(35,39);(10,12,14,22,15,23,29,41)(11,16,18,20,19,31,27,36)(13,21,28,40,26,39,25,38)(17,32,34,37,24,30,33,35);(10,13,14,28,15,26,29,25)(11,17,18,34,19,24,27,33)(12,21,22,40,23,39,41,38)(16,30,20,35,31,32,36,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4338);