Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×C5×#1708 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(27,28,29,30,31);(4,7)(5,9)(8,10)(11,12)(13,19)(14,22)(15,17)(16,20)(18,23)(21,25)(24,26);(11,13,14,20,15,21,23,26)(12,16,18,19,17,24,22,25);(3,4,5,7,6,8,9,10)(12,17)(16,24)(18,22)(19,25)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12583);