Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15,19,21)(16,18,23,28)(17,24,25,22)(20,26,29,27);(7,10)(11,12)(14,16,18,21,19,23,28,15)(17,22,26,29,25,24,27,20);(7,11)(9,13)(10,12)(14,17)(15,20)(16,22)(18,26)(19,25)(21,29)(23,24)(27,28);(17,25)(20,29)(22,24)(26,27)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12187);