Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2206 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 135 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)(10,19)(11,15)(13,16)(14,17)(18,21)(20,22);(3,4,5,6);(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20)58 conjugacy classes — view the full character table.
| maximal subgroup of | #136893, #136898, #136927, … |
| maximal normal subgroup of | #136893, #136898, #136927, … |
| maximal quotient of | C2×#12416, C2×#12424, C2×#12445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26561);