Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #481 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1919 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,17,10,16,13,27)(8,18,21,35,20,14,11,25)(9,22,23,29,24,37,28,34)(15,30,31,36,33,26,32,19);(6,8,12,21,10,20,13,11)(7,14,17,25,16,18,27,35)(9,19,23,30,24,36,28,26)(15,29,31,37,33,34,32,22);(6,9,12,23,10,24,13,28)(7,15,27,32,16,33,17,31)(8,19,21,30,20,36,11,26)(14,29,35,22,18,34,25,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6103);