Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2399 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| socle, prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1407 conjugacy classes of subgroups; only the 639 normal subgroups are shown.
This group has 1407 conjugacy classes of subgroups — too many to draw. Only its 639 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 — 5 generators on 22 points:
(1,2);(7,8,11,14)(9,13,10,12)(15,16,19,22)(17,21,18,20);(3,4)(5,6)(7,9,11,10)(8,12,14,13)(15,17,19,18)(16,20,22,21);(4,6)(7,9,11,10)(8,13,14,12)(15,18,19,17)(16,21,22,20);(7,10,11,9)(8,12,14,13)88 conjugacy classes — view the full character table.
| maximal subgroup of | #111497, #115162, #121997, … |
| maximal normal subgroup of | #111497, #115162, #121997, … |
| maximal quotient of | C2×#13848, C2×#13850, C2×#13856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27534);