Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C22×C5×#490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C22×C5×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
2830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(21,22,23,24,25);(3,4);(5,6)(7,10)(8,9)(11,12)(14,19)(15,20)(16,17);(6,9)(10,12)(13,14,16,19,18,20,17,15);(5,7,8,11)(6,10,9,12)(13,15,16,14,18,19,17,20)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53595);