Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #127 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1955 |
| prime radical | C5×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
438 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)(6,7,11,17,12,16,28,32,13,8,14,20,10,19,27,35)(9,21,25,36,26,18,33,34,23,22,29,37,24,15,30,31);(6,8,11,20,12,19,28,35,13,7,14,17,10,16,27,32)(9,22,29,36,26,15,30,34,23,21,25,37,24,18,33,31);(6,9,10,24,13,23,12,26)(7,15,16,21,8,18,19,22)(11,25,27,30,14,29,28,33)(17,31,32,36,20,34,35,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6188);