Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10568 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
560 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)(6,7,10,12)(8,13,9,11)(14,15,17,20)(16,21,18,19)(22,23)(24,26)(25,27)(28,29);(6,8,10,9)(7,11,12,13)(14,16,17,18)(15,19,20,21)(22,24)(23,26)(25,28)(27,29);(6,9,10,8)(7,11,12,13)(15,19)(16,18)(20,21)(26,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3556);