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) = C25 |
| direct product | C5×#22669 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C22×C5×D4 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×D4 ◁ G | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,14,17,15,18,13,23)(12,19,22,24,20,25,21,16);(6,7)(8,9)(12,20)(16,24)(19,25)(21,22);(10,12,13,21,15,20,14,22)(11,16,17,19,18,24,23,25);(7,9)(12,20)(16,24)(19,25)(21,22)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15657);