Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | Aut(G) = C2×#72308187 (order 4096) · Inn(G) = #447 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 41 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:
(5,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17);(1,3,4,7)(2,5,6,8)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg34 conjugacy classes — view the full character table.
| maximal subgroup of | #123061, #123063, #123067, … |
| maximal normal subgroup of | #123061, #123063, #123067, … |
| maximal quotient of | #123132, #123135, #123138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5147);