Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#1437 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C26 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2615 conjugacy classes of subgroups; only the 727 normal subgroups are shown.
This group has 2615 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) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,6)(7,8,10,9)(12,13)(15,16,18,17);(11,12,14,13)(15,16,18,17);(3,4,5,6)(7,9,10,8)(11,13,14,12)(15,17,18,16)112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6748);