Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | Aut(G) = C24×#525 (order 1024) · Inn(G) = #117 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,14,8,13,16,20,10,7,11,15,17,12,18,19);(5,7,9,14,8,12,16,20,10,6,11,15,17,13,18,19);(2,4)(6,12,7,13)(9,11)(14,20,15,19)(16,18)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cg,d^b=dh,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13128, #494844, #494845, … |
| maximal normal subgroup of | C2×#13128, #494844, #494845, … |
| maximal quotient of | #140133, #140143, #140151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6116);