Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#20398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,19)(13,17)(14,23)(15,18)(16,21)(20,25)(22,24);(10,12,14,22,15,21,13,20)(11,16,17,24,18,19,23,25);(6,8,9,7)(10,11)(12,16)(13,23)(14,17)(15,18)(19,21)(20,25)(22,24);(10,13,15,14)(11,17,18,23)(12,20,21,22)(16,24,19,25)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13386);