Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#1634 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#555 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
1074 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)(11,12,16,19)(13,20,22,25)(14,18,15,23)(17,24,26,21);(5,9)(7,10)(11,13,16,22)(12,17,19,26)(14,21,15,24)(18,25,23,20);(3,4,6,8)(5,10,9,7)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11497);