Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C2×#10586342 (order 1024) · Inn(G) = #124 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1924 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,6,11,7,12,23,14,5,10,13,17,20,26,28,30)(3,4,9,18,15,19,25,29,16,8,21,22,27,24,31,32);(1,3,13,9,7,15,28,25,5,16,6,21,20,27,23,31)(2,8,11,18,12,24,14,29,10,4,17,22,26,19,30,32);(1,4,5,8)(2,9)(3,14)(6,22,13,18)(7,19,20,24)(10,21)(11,27)(12,25)(15,17)(16,30)(23,32,28,29)(26,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13129, #494848, #494957, … |
| maximal normal subgroup of | C2×#13129, #494848, #494957, … |
| maximal quotient of | #140133, #140143, #140151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6117);