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) = C26 |
| direct product | C2×#2186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 599 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);(3,4)(5,6)(8,13)(9,12)(10,17)(14,22)(15,21)(16,19)(18,20);(7,8)(9,14)(10,18)(11,15)(12,20)(13,17)(16,21)(19,22);(4,6)(7,9,11,12)(8,14,15,20)(10,16,19,17)(13,18,21,22)58 conjugacy classes — view the full character table.
| maximal subgroup of | #138913, #138919, C4×#2186, … |
| maximal normal subgroup of | #138913, #138919, C4×#2186, … |
| maximal quotient of | C2×#12104, C2×#12280, C2×#12160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26502);