Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11046 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C25 | C23×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
1178 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)(7,9)(10,11)(12,15)(13,20,21,16)(14,17)(18,22,25,24)(19,23);(6,7,8,9)(10,12)(11,15)(13,18)(14,19)(16,24)(17,23)(20,22)(21,25);(10,13)(11,16)(12,18)(14,21)(15,22)(17,20)(19,25)(23,24)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4034);