Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
844 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)(6,7,9,8)(10,11,15,20,16,21,14,19)(12,22,25,17,26,32,24,33)(13,18,29,36,30,37,28,35)(23,34,38,31,39,41,27,40);(6,8,9,7)(10,12)(11,17)(13,27)(14,25)(15,24)(16,26)(18,34)(19,32)(20,22)(21,33)(23,28)(29,39)(30,38)(31,35)(36,40)(37,41);(10,12)(11,17)(13,23)(14,24)(15,25)(16,26)(18,31)(19,22)(20,32)(21,33)(27,28)(29,38)(30,39)(34,35)(36,41)(37,40);(10,13,15,29,16,30,14,28)(11,18,20,36,21,37,19,35)(12,23,25,38,26,39,24,27)(17,31,32,41,33,40,22,34)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16033);