Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20954 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,21,16,19,14,20)(12,22,26,17,25,33,24,32)(13,27,29,34,30,18,28,35)(23,36,39,40,38,41,37,31);(6,7,8,9)(10,12)(11,17)(13,23)(14,26)(15,24)(16,25)(18,31)(19,32)(20,33)(21,22)(27,40)(28,39)(29,37)(30,38)(34,36)(35,41);(10,13,16,30)(11,18,19,27)(12,23,25,38)(14,29,15,28)(17,31,32,40)(20,35,21,34)(22,36,33,41)(24,39,26,37);(11,19)(17,32)(18,27)(20,21)(22,33)(31,40)(34,35)(36,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13942);