Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8)(9,12)(10,13)(11,19)(14,18)(15,17)(16,20)(21,22);(3,4,5,6)(8,12)(13,20)(14,21)(15,19);(8,13)(10,18)(11,17)(12,20)(14,15)(16,22)(19,21)58 conjugacy classes — view the full character table.
| maximal subgroup of | #138856, #138862, C4×#2210, … |
| maximal normal subgroup of | #138856, #138862, C4×#2210, … |
| maximal quotient of | C2×#12418, C2×#12422, C2×#12428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26563);