Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12750 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1786 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1786 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,6,10,7,4,18,11)(3,12,16,26,17,30,15,29)(5,20,22,9,23,27,32,19)(8,21,25,14,13,31,24,28);(1,3)(2,8)(4,13)(5,21)(6,15)(7,17)(9,26)(10,24)(11,25)(12,27)(14,32)(16,18)(19,29)(20,30)(22,28)(23,31);(1,4)(2,7)(3,13)(5,19)(6,10)(8,17)(9,23)(11,18)(12,28)(14,30)(15,24)(16,25)(20,32)(21,29)(22,27)(26,31)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fh,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12752, #493558, #495737, … |
| maximal normal subgroup of | C2×#12752, #493558, #495737, … |
| maximal quotient of | #129522, #129523, #129565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5740);