Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C5×#8316 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C2×C5×H(5,𝔽2) | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1118 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 3 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,16,18,17,19,20,21);(6,8)(7,10)(9,13)(11,12);(6,7)(8,10)(9,11)(12,13)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1304);