Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C8 |
| direct product | C5×#19745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
994 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(15,18)(16,20)(19,21);(6,7,8,10,9,11,12,13);(14,15,16,18,17,19,20,21);(7,11)(10,13)(15,19)(18,21)200 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#19745, C52×#19745, C2×C5×#19745, … |
| maximal normal subgroup of | D5×#19745, C52×#19745, C2×C5×#19745, … |
| maximal quotient of | C52×#19745, C2×C5×#19745, C11⋊C5×#19745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12733);