Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#1785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1963 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 1963 conjugacy classes of subgroups — too many to draw. Only its 311 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)(7,8,11,16,12,15,10,14)(9,17,18,22,19,13,20,21);(7,9)(8,13)(10,18)(11,20)(12,19)(14,21)(15,17)(16,22);(4,6)(7,10,12,11)(8,14,15,16)(9,18,19,20)(13,21,17,22)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105826, #106150, #106546, … |
| maximal normal subgroup of | #105826, #106150, #106546, … |
| maximal quotient of | C2×#8300, C2×#8301, #184747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13578);