Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #383 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 183 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)(5,13,16,17)(6,9)(7,11)(8,12)(10,15,18,14);(4,9,12,11)(10,17,18,13);(3,5,6,14,8,16,7,15)(4,10,11,13,12,18,9,17)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7225, #127312, C2×#7226, … |
| maximal normal subgroup of | C2×#7225, #127312, C2×#7226, … |
| maximal quotient of | C2×#8800, C2×#8801, #191286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13763);