Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C26 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
1395 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 1395 conjugacy classes of subgroups — too many to draw. Only its 223 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 — 4 generators on 18 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,13,16,14,15,17,18);(3,5)(4,7)(6,10)(8,9);(12,15)(16,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105811, #105814, #106135, … |
| maximal normal subgroup of | #105811, #105814, #106135, … |
| maximal quotient of | C2×#7606, C2×#7609, C2×#7612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10342);