Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| prime radical | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×Q8 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,15,19,16,20,13,17)(12,21,23,14,18,22,25,24);(10,12,22,19,16,18,21,17)(11,15,25,14,20,13,23,24);(6,7,8,9)(10,13,16,15)(11,17,20,19)(12,23,18,25)(14,22,24,21)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6216);