Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#10699 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C42 |
| commutator | C4 | C43×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #376 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
842 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)(12,22,25,32)(13,18)(14,19)(15,21)(16,20)(17,24,31,26)(23,36,38,41)(27,33)(28,35)(29,34)(30,37,40,39);(7,9)(10,12)(11,17,20,31)(13,23)(14,24)(15,26)(16,25)(18,30,34,40)(19,22,21,32)(27,37)(28,39)(29,38)(33,36,35,41);(10,13,15,28,16,29,14,27)(11,18,21,35,20,34,19,33)(12,23,26,39,25,38,24,37)(17,30,32,41,31,40,22,36)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3687);