Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C23×#123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C23 |
| commutator, Frattini, cosocle | C22 | C26 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
7575 conjugacy classes of subgroups; only the 3439 normal subgroups are shown.
This group has 7575 conjugacy classes of subgroups — too many to draw. Only its 3439 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 18 points:
(3,4);(5,6);(1,2)(3,4)(5,6);(8,10)(12,15)(13,17);(11,12)(13,16)(14,15)(17,18);(7,8)(9,10)(11,13,14,17)(12,16,15,18)112 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×A4×#123, #123×AGL(1,𝔽8) |
| maximal subgroup of | #154299, #154328, #154672, … |
| maximal normal subgroup of | #154299, #154328, #154672, … |
| maximal quotient of | C23×#381, C23×#385, C23×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55646);