Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,11,18,15,19,20,9,12,17,16,10,7,13,14);(1,2,3,4)(5,7)(6,10)(8,16)(9,15)(11,17)(12,18)(13,14)(19,20);(8,17)(11,16)(13,19)(14,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131322, #131324, #131327, … |
| maximal normal subgroup of | #131322, #131324, #131327, … |
| maximal quotient of | #136609, #136637, #136883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5987);