Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×Q8 ◁ C2×C5×2−1+4 ◁ G | |
1220 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)(8,18)(9,14)(10,15)(11,16,25,27)(12,17)(13,20,28,24)(19,29)(21,30,35,37)(22,31)(23,32)(26,33,36,34);(6,8,12,10)(7,13,17,28)(9,19,22,23)(11,24,25,20)(14,26,31,36)(15,27,18,16)(21,34,35,33)(29,30,32,37);(6,9,12,22)(7,14,17,31)(8,19,10,23)(11,21,25,35)(13,26,28,36)(15,32,18,29)(16,30,27,37)(20,33,24,34);(6,10,12,8)(7,15,17,18)(9,19,22,23)(11,24,25,20)(13,27,28,16)(14,29,31,32)(21,33,35,34)(26,30,36,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26350);