Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12606 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C42 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#97 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
466 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)(12,22,24,31)(13,19)(14,28)(15,21)(16,18)(17,23,32,26)(20,29)(25,39,38,36)(27,35)(30,34)(33,40,41,37);(7,9)(10,12,14,25,15,26,30,40,16,24,29,38,13,23,27,37)(11,17,20,33,21,31,35,39,18,32,28,41,19,22,34,36);(11,18)(12,23,24,26)(14,29)(17,31,32,22)(19,21)(25,40,38,37)(27,30)(33,36,41,39)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5594);