Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8189 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
420 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)(24,27)(25,29)(26,28)(30,34)(31,33)(32,35)(36,37);(7,12,15,14)(13,19,21,20)(22,24,25,31,26,30,32,36)(23,27,29,33,28,34,35,37);(6,8,10,17,11,18,9,16)(7,13,14,20,15,21,12,19)(23,28)(24,30)(29,35)(31,36)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1177);