Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #327 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,6,11,8,12,7,16)(5,9,13,10,15,18,14,17);(4,9)(5,13)(6,7)(10,11)(12,18)(14,15)(16,17)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1341, #128804, #128832, … |
| maximal normal subgroup of | C4×#1341, #128804, #128832, … |
| maximal quotient of | C2×#7084, C2×#7085, #122616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4674);