Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| socle, prime radical | C25 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2879 conjugacy classes of subgroups; only the 2063 normal subgroups are shown.
This group has 2879 conjugacy classes of subgroups — too many to draw. Only its 2063 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) = 256 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(10,12)(13,14,15,16);(9,10,11,12)160 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×A4×C4⋊C4 |
| maximal subgroup of | #105512, #105532, #105569, … |
| maximal normal subgroup of | #105512, #105532, #105569, … |
| maximal quotient of | C22×C4×#95, C2×C42×C4⋊C4, C22×#1434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26977);