Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C5×#1757 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
1474 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);(3,4,7,11,8,12,19,25)(5,13,16,27,15,9,20,22)(6,10,17,23,18,24,30,33)(14,26,29,34,28,21,31,32);(3,5)(4,9,12,13)(6,14)(7,16)(8,15)(10,21,24,26)(11,22,25,27)(17,29)(18,28)(19,20)(23,32,33,34)(30,31);(3,6,7,17,8,18,19,30)(4,10,11,23,12,24,25,33)(5,14,16,29,15,28,20,31)(9,21,22,32,13,26,27,34)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13388);