Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C22×#343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1247 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1247 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2);(3,4);(9,10,11,14,12,13,15,16);(5,6,7,8)(10,13)(11,15)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#343 |
| maximal subgroup of | #123439, #123455, #123469, … |
| maximal normal subgroup of | #123439, #123455, #123469, … |
| maximal quotient of | C22×#1280, C2×#7908, C2×#7909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13344);