Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17840 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C2×C5×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,11)(9,12)(13,18)(14,19)(15,21)(17,20)(22,23,26,29,27,30,25,35)(24,31,34,36,32,37,33,28);(6,7,12,19,10,14,9,11)(8,15,20,18,16,13,17,21)(22,24,26,34,27,32,25,33)(23,28,35,37,30,36,29,31);(6,8)(7,13)(9,17)(10,16)(11,18)(12,20)(14,15)(19,21)(24,32)(28,36)(31,37)(33,34);(8,16)(13,15)(17,20)(18,21)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10828);