Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#2112 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22×C8 |
| prime radical | C25 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 287 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) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,9,13,10,14,15,20,11,12,16,18,17,19,21,22);(4,6);(8,12)(13,18)(14,19)(20,22)112 conjugacy classes — view the full character table.
| maximal subgroup of | #126898, #126899, C2×#7184, … |
| maximal normal subgroup of | #126898, #126899, C2×#7184, … |
| maximal quotient of | C2×#11418, C2×#11440, C2×#11460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26319);