Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#12660 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | #432 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
330 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)(13,25)(14,19)(15,17)(16,22)(18,26)(21,29)(23,30)(24,33)(27,39)(28,31)(32,35)(34,37)(36,41)(38,40);(7,9)(10,12,13,23,14,24,27,38,15,22,26,37,28,21,35,36)(11,16,18,30,19,29,32,40,17,20,25,34,31,33,39,41);(11,17)(12,21,22,24)(13,26)(16,29,20,33)(19,31)(23,38,37,36)(27,35)(30,41,34,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5648);