Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
713 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 713 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,19,16,20,14,18)(13,17,21,24,22,25,23,26);(3,5,6,10)(4,7,8,9)(11,13)(12,17)(14,23)(15,21)(16,22)(18,26)(19,24)(20,25);(11,14,16,15)(12,18,20,19)(13,21,22,23)(17,24,25,26)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105766, #106078, #107199, … |
| maximal normal subgroup of | #105766, #106078, #107199, … |
| maximal quotient of | C2×#7665, C2×#7668, C2×#7671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10366);