Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#9588 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #388 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,11)(8,15)(9,12)(13,21)(14,19)(16,18)(22,23,25,28,27,31,26,30)(24,32,34,37,33,36,35,29);(6,7,12,19,10,14,9,11)(8,16,20,18,15,13,17,21)(23,28,31,30)(24,33)(29,36,37,32)(34,35);(6,8,9,17,10,15,12,20)(7,13,11,18,14,16,19,21)(22,24,25,34,27,33,26,35)(23,29,30,36,31,37,28,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2576);