Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4)(5,13)(6,10)(7,20,21,11)(8,12)(9,15,23,30)(14,27)(16,28)(17,31)(18,33,34,25)(19,26)(22,29,24,32);(3,5,8,16)(4,9,12,23)(6,17,19,14)(7,15,21,30)(10,24,26,22)(11,13,20,28)(18,32,34,29)(25,31,33,27);(3,6,8,19)(4,10,12,26)(5,14,16,17)(7,18,21,34)(9,22,23,24)(11,25,20,33)(13,27,28,31)(15,29,30,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26334);