Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12756 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,12,17,9,14)(8,18,22,33,21,34,19,35)(10,24,23,31,25,15,26,30)(13,27,28,20,29,36,32,37);(6,8)(7,13)(9,19)(10,20)(11,22)(12,21)(14,28)(15,18)(16,32)(17,29)(23,36)(24,34)(25,37)(26,27)(30,35)(31,33);(6,9,12,11)(7,14,17,16)(8,19,21,22)(10,23,25,26)(13,28,29,32)(15,31,24,30)(18,33,34,35)(20,36,37,27)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5744);