Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
332 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)(10,11,16,22,15,20,13,31,17,23,32,19,14,21,25,38)(12,24,29,40,28,37,26,34,30,35,41,39,27,18,33,36);(10,12,25,33,14,27,32,41,17,30,13,26,15,28,16,29)(11,18,31,40,21,35,22,36,23,37,38,39,20,24,19,34);(7,9)(10,13,14,16,17,25,15,32)(11,19,20,38,23,22,21,31)(12,26,27,29,30,33,28,41)(18,34,24,39,37,36,35,40)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6112);