Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#2372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1579 conjugacy classes of subgroups; only the 775 normal subgroups are shown.
This group has 1579 conjugacy classes of subgroups — too many to draw. Only its 775 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 — 5 generators on 22 points:
(1,2);(3,4,6,5)(8,11)(12,14)(15,16,19,18)(17,21,22,20);(3,5,6,4)(16,18)(20,21);(7,8)(9,12)(10,11)(13,14)(15,17,19,22)(16,20,18,21);(7,9,10,13)(8,12,11,14)(15,18,19,16)(17,21,22,20)100 conjugacy classes — view the full character table.
| maximal subgroup of | #107287, #146629, #146640, … |
| maximal normal subgroup of | #107287, #146629, #146640, … |
| maximal quotient of | C2×#13805, C2×#13816, C4×#2372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27135);