Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C4×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C2×D8 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6);(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)88 conjugacy classes — view the full character table.
| maximal subgroup of | #136685, #136689, #136719, … |
| maximal normal subgroup of | #136685, #136689, #136719, … |
| maximal quotient of | C2×C8×D16, C2×#12069, C2×#12081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26432);