Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#24173 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,10)(11,13)(14,15,18,22)(16,23,25,28)(17,27,19,21)(20,24,29,26);(7,10)(11,12)(14,15)(16,23)(17,21)(18,22)(19,27)(20,24)(25,28)(26,29);(14,16,18,25)(15,20,22,29)(17,24,19,26)(21,28,27,23);(6,8,9,13)(7,11,10,12)(14,17,18,19)(15,21,22,27)(16,24,25,26)(20,28,29,23)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17161);