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) = C210×C4 |
| direct product | C22×C4×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| socle | C25 | C23 |
| prime radical | C26 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4767 conjugacy classes of subgroups; only the 2063 normal subgroups are shown.
This group has 4767 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) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(9,10)(11,12)(13,14,15,16);(10,12)160 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×A4×C22⋊C4 |
| maximal subgroup of | #105511, #105528, #105568, … |
| maximal normal subgroup of | #105511, #105528, #105568, … |
| maximal quotient of | C22×C4×#95, C2×C42×C22⋊C4, C22×C22⋊#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26976);