Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C22⋊C42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C26 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2127 conjugacy classes of subgroups; only the 471 normal subgroups are shown.
This group has 2127 conjugacy classes of subgroups — too many to draw. Only its 471 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) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(9,10,11,12)(14,16);(13,14)(15,16);(2,4)(5,6,7,8)(9,12,11,10);(1,2)(3,4)(5,6)(7,8)100 conjugacy classes — view the full character table.
| maximal subgroup of | #110086, #139622, C22⋊C4×#99, … |
| maximal normal subgroup of | #110086, #139622, C22⋊C4×#99, … |
| maximal quotient of | C22⋊C4×#95, #153727, #153749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6786);