Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3486 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×Q8 ◁ C2×C5×2−1+4 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,13)(9,14)(10,22,23,12)(11,15)(16,26)(17,31,32,24)(18,27)(19,28)(20,34,35,25)(21,29)(30,37,33,36);(7,12)(13,24)(14,25)(15,22)(26,36)(27,31)(28,37)(29,34);(6,8,11,18)(7,13,15,27)(9,19,21,16)(10,17,23,32)(12,24,22,31)(14,28,29,26)(20,33,35,30)(25,37,34,36);(6,9,11,21)(7,14,15,29)(8,16,18,19)(10,20,23,35)(12,25,22,34)(13,26,27,28)(17,30,32,33)(24,36,31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26337);