Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#2503 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1459 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1459 conjugacy classes of subgroups — too many to draw. Only its 567 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 — 5 generators on 22 points:
(1,2);(4,6)(8,10)(9,13)(15,16,19,21)(17,18,20,22);(7,8,11,10)(9,12,13,14)(15,17)(16,18)(19,20)(21,22);(3,4,5,6)(7,9)(8,12)(10,14)(11,13)(15,18)(16,20)(17,21)(19,22);(7,10,11,8)(9,14,13,12)76 conjugacy classes — view the full character table.
| maximal subgroup of | #98506, #111898, #112548, … |
| maximal normal subgroup of | #98506, #111898, #112548, … |
| maximal quotient of | C2×#13881, C2×#13904, C2×#13972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27761);