Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
338 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 338 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:
(1,2,5,8,6,3,4,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23);(1,3,6,2)(4,8,5,7)(9,11,13,21,14,22,12,20)(10,15,16,23,18,24,17,19);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #105437, #105438, #105439, … |
| maximal normal subgroup of | #105437, #105438, #105439, … |
| maximal quotient of | #95371, #95374, #95375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1080);