Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1778 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
523 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 523 conjugacy classes of subgroups — too many to draw. Only its 351 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);(3,4,6,5)(7,8,11,16,12,14,10,15)(9,17,19,13,20,22,18,21);(7,9,10,19,12,20,11,18)(8,13,15,22,14,21,16,17);(3,5,6,4)(8,14)(13,21)(15,16)(17,22)88 conjugacy classes — view the full character table.
| maximal subgroup of | #109536, #109537, #116958, … |
| maximal normal subgroup of | #109536, #109537, #116958, … |
| maximal quotient of | C2×#8275, #183364, C2×#8277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13445);