Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#8953 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,9,12,10,8,11,13)(14,15,18,22,19,23,17,21)(16,24,26,20,27,28,25,29);(6,8,9,13,10,7,11,12)(14,16)(15,20)(17,26)(18,25)(19,27)(21,28)(22,24)(23,29);(14,16)(15,20)(17,25)(18,26)(19,27)(21,24)(22,28)(23,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1941);