Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 247 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)(8,13)(9,16)(10,11)(14,21)(15,19)(17,18);(3,5,6,4)(7,8)(9,14)(10,19)(11,13)(12,15)(16,20)(17,22)(18,21);(7,9,11,16,12,18,10,17)(8,14,13,20,15,21,19,22)64 conjugacy classes — view the full character table.
| maximal subgroup of | #147202, #147207, #147232, … |
| maximal normal subgroup of | #147202, #147207, #147232, … |
| maximal quotient of | C2×#9607, C2×#9609, #203716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14167);