Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 5505024 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C23×#99 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22⋊C4 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
3951 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 3951 conjugacy classes of subgroups — too many to draw. Only its 1039 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) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(3,4);(5,6);(7,8)(9,13,14,11)(10,12);(7,9)(8,11)(10,14)(12,13)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53054);