Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#3370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C2×C5×#459 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×Q8 ◁ G | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(4,7)(8,9)(11,12)(13,20)(14,18)(15,17)(16,21)(19,23)(22,24)(25,26);(3,4,6,7)(5,9,10,8)(11,13,14,21)(12,16,18,20)(15,22,23,26)(17,25,19,24);(12,17)(13,21)(15,23)(16,24)(18,19)(20,25);(3,5,6,10)(4,8,7,9)(13,21)(16,20)(22,26)(24,25)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 54300);