Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#2313 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1447 conjugacy classes of subgroups; only the 783 normal subgroups are shown.
This group has 1447 conjugacy classes of subgroups — too many to draw. Only its 783 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);(3,4,5,6)(7,8,11,9)(10,13,14,12)(17,21)(19,22);(8,9)(12,13)(15,16,18,20)(17,19,21,22);(7,9,11,8)(10,13,14,12)(15,17)(16,19)(18,21)(20,22);(7,10)(8,12)(9,13)(11,14)88 conjugacy classes — view the full character table.
| maximal subgroup of | #105607, #105812, #105818, … |
| maximal normal subgroup of | #105607, #105812, #105818, … |
| maximal quotient of | C2×#13776, C2×#13778, C2×#13787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27034);