Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×#41 ◁ G | |
5058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,18)(12,16,19,23)(13,15,21,17)(20,22,25,24);(11,15)(13,21)(16,22)(17,18)(20,25)(23,24);(7,9);(10,12)(11,16)(13,20)(14,19)(15,22)(17,24)(18,23)(21,25);(12,19)(16,23)(20,25)(22,24)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53361);