Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,12)(9,24)(10,27)(11,17)(13,19)(14,22)(15,25)(16,23)(20,35)(21,31)(26,30)(28,33)(29,36)(32,34);(6,12)(7,19)(8,22)(9,25)(10,23)(14,16)(15,27)(17,24)(18,28)(20,36)(21,26)(29,32)(31,35)(33,34);(2,4)(5,7,8,19,23,33,25,36,11,21,9,26,10,20,22,34)(6,13,14,18,27,32,24,35,17,30,15,31,16,29,12,28)Polycyclic presentation:
2^8|c^2=d,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93814, #93816, #151288, … |
| maximal normal subgroup of | #93814, #93816, #151288, … |
| maximal quotient of | #145359, #145391, #145450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6427);