Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,19,23)(16,24,27,28)(17,22,18,21)(20,26,29,25);(6,7,9,11)(8,13,12,10)(14,16,19,27)(15,20,23,29)(17,26,18,25)(21,28,22,24);(15,21)(16,25)(17,18)(22,23)(24,28)(26,27)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3174);