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) = C29 |
| direct product | C2×#2551 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1591 conjugacy classes of subgroups; only the 727 normal subgroups are shown.
This group has 1591 conjugacy classes of subgroups — too many to draw. Only its 727 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,5)(11,12,15,18)(13,17,14,16);(3,4,6,5)(7,8,9,10)(11,13,15,14)(12,16,18,17);(7,8)(9,10)(11,13,15,14)(12,16,18,17);(3,5,6,4)(11,14,15,13)(12,17,18,16)76 conjugacy classes — view the full character table.
| maximal subgroup of | #98536, #122060, #122092, … |
| maximal normal subgroup of | #98536, #122060, #122092, … |
| maximal quotient of | C2×#13910, C2×#14005, C2×#14053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27912);