Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #72313686 (order 4096) · Inn(G) = #481 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1824 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,7,11,6,22)(3,12,15,26,13,28,16,31)(4,17,20,25,18,32,21,9)(8,19,24,14,23,30,27,29);(4,18)(9,25)(12,28)(14,29)(17,32)(19,30)(20,21)(26,31);(1,3,6,15,7,13,5,16)(2,8,10,27,11,23,22,24)(4,19,21,14,18,30,20,29)(9,26,17,12,25,31,32,28)Polycyclic presentation:
2^8|a^2=f,c^2=dg,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #137211, #137214, #137217, … |
| maximal normal subgroup of | #137211, #137214, #137217, … |
| maximal quotient of | #135268, #135288, #135347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5864);