Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8246 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C23×C5 | #117 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
376 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)(6,7,10,14,11,15,9,12)(8,13,17,20,18,21,16,19)(22,23,25,28)(24,30,31,27)(26,34,33,29)(32,35,37,36);(7,12,15,14)(13,19,21,20)(22,24,26,32,25,31,33,37)(23,27,29,35,28,30,34,36);(6,8,10,17,11,18,9,16)(7,13,14,20,15,21,12,19)(24,31)(27,30)(32,37)(35,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1234);