Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C2×C4×C16 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4);(5,6)(7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(6,8)160 conjugacy classes — view the full character table.
| maximal subgroup of | #93274, #93277, #93278, … |
| maximal normal subgroup of | #93274, #93277, #93278, … |
| maximal quotient of | C8×C22⋊C16, #128251, #128271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5586);