Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#544 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3815 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 3815 conjugacy classes of subgroups — too many to draw. Only its 3183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(3,4);(1,2)(3,4);(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17);(5,6,8,10)(7,9,12,11);(5,6)(7,9)(8,10)(11,12)(13,15,16,20)(14,17,18,19);(5,7)(6,9)(8,12)(10,11)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#544 |
| maximal subgroup of | C2×#8406, #185936, #185975, … |
| maximal normal subgroup of | C2×#8406, #185936, #185975, … |
| maximal quotient of | C22×#2316, C22×#2317, C22×#2329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55664);