Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#18143 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1170 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)(10,13)(14,15)(16,22)(17,25)(18,20)(19,23)(21,27)(24,29)(26,28);(6,7)(8,10)(9,11)(12,13)(14,16,18,24)(15,19,20,28)(17,26,27,23)(21,29,25,22);(6,8,9,12)(7,10,11,13);(15,20)(19,28)(21,25)(22,29)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11131);