Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C2×C8 |
| direct product | C5×#8266 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C8 | D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C4⋊C4 |
| prime radical | C23×C5 | #117 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(15,19,21,24)(20,26,27,28);(6,7,8,10,9,11,12,13)(14,15)(16,20)(17,24)(18,21)(19,25)(22,28)(23,27)(26,29);(14,16,17,22,18,23,25,29)(15,20,24,28,21,27,19,26)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1254);