Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#1375 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C42 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C23 | #97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 315 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) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,10,15,11,13,19,20,12,16,18,22,9,14,17,21);(4,6)(8,13,16,14)(10,18)(15,21,22,20)(17,19)88 conjugacy classes — view the full character table.
| maximal subgroup of | #93378, #93379, #93392, … |
| maximal normal subgroup of | #93378, #93379, #93392, … |
| maximal quotient of | C2×#7137, #126550, #126562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5534);