Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(2,5)(3,7)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(2,5)(6,8)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=h,f^2=gh,b^a=bd,c^a=ch,d^a=dg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105898, #105899, #105998, … |
| maximal normal subgroup of | #105898, #105899, #105998, … |
| maximal quotient of | #94918, #94921, #94930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 923);