Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13188 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1948 |
| prime radical | C23×C5 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
280 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)(6,7,11,18,13,19,12,17)(8,20,24,34,25,31,23,37)(9,26,27,15,28,16,10,29)(14,30,32,22,33,36,35,21);(6,8)(7,14,19,33)(9,21)(10,30)(11,23)(12,24)(13,25)(15,31,29,20)(16,34,26,37)(17,32,18,35)(22,28)(27,36);(6,9,11,27,13,28,12,10)(7,15,17,26,19,29,18,16)(8,21,23,36,25,22,24,30)(14,31,32,37,33,20,35,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6176);