Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#3123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
2210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4)(5,8)(6,7)(9,10)(11,12,14,18)(13,20,21,16)(15,17,23,19)(22,25,26,24);(4,7)(8,10)(11,13)(12,16)(14,21)(15,22)(17,24)(18,20)(19,25)(23,26);(12,17)(15,23)(16,24)(18,19)(20,25)(22,26);(3,5,6,9)(4,8,7,10)(13,21)(16,20)(22,26)(24,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53540);