Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×#1342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #327 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,7,12,9,13,8,24)(5,14,16,28,18,31,17,34)(6,19,21,11,23,29,22,30)(10,20,25,33,27,32,26,15);(3,5,9,18)(4,10,13,27)(6,20,23,32)(7,17,8,16)(11,28,30,34)(12,25,24,26)(14,29,31,19)(15,22,33,21)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4675);