Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12574 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ C23×C5×D4 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(10,11)(12,18)(13,21,22,16)(14,17)(15,20,23,25)(19,24);(6,7,8,9)(10,12)(11,15)(13,20)(14,19)(16,18)(17,23)(21,24)(22,25);(12,19)(15,23)(18,24)(20,25)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5562);