Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#32887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×2−1+4 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C5×#41 ◁ G | |
3592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,13)(14,15,18,20)(16,22,24,29)(17,25,27,21)(19,26,28,23);(14,16,18,24)(15,19,20,28)(17,26,27,23)(21,29,25,22);(7,9)(14,15)(16,19)(17,21)(18,20)(22,23)(24,28)(25,27)(26,29);(11,13)(15,20)(19,28)(21,25)(22,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25875);