Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2087 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
527 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 223 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) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8)(9,14)(10,12)(11,13)(15,16,19,20,18,22,17,21);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,18,19)(16,20,22,21);(4,6)(7,10,11,9)(8,12,13,14)(16,21)(17,19)(20,22)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7081, #122599, C2×#7089, … |
| maximal normal subgroup of | C2×#7081, #122599, C2×#7089, … |
| maximal quotient of | C2×#10650, C2×#10656, C2×#10652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16653);