Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1831 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 367 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 367 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);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16,17,19,18,20,21,22);(3,4,6,5)(8,11)(12,14);(3,5,6,4)(7,9,10,13)(8,12,11,14)112 conjugacy classes — view the full character table.
| maximal subgroup of | #94923, #94927, #94935, … |
| maximal normal subgroup of | #94923, #94927, #94935, … |
| maximal quotient of | C2×#8780, C2×#8781, #191136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13738);