Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#3003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
3557 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 3557 conjugacy classes of subgroups — too many to draw. Only its 583 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14);(8,11)(9,14)(12,13);(4,6);(8,12)(11,13)64 conjugacy classes — view the full character table.
| maximal subgroup of | #114565, #114862, #114979, … |
| maximal normal subgroup of | #114565, #114862, #114979, … |
| maximal quotient of | C2×#18132, C2×#18137, C2×#18142, … |
| automorphism group of | #334, #335, C3×#106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53342);