Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 53 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)(5,6,10,11,12,16,9,15)(7,17,20,13,21,30,19,31)(8,22,24,25,27,28,26,14)(18,34,35,23,32,29,36,33);(5,7,12,21)(6,13,16,31)(8,23,27,33)(9,20,10,19)(11,17,15,30)(14,32,25,18)(22,35,28,36)(24,34,26,29);(2,4)(5,8,9,24,12,27,10,26)(6,14,15,22,16,25,11,28)(7,18,19,35,21,32,20,36)(13,29,17,33,31,34,30,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=d,d^2=h,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,g^b=gh,e^c=eh,f^c=fgh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126284, #126285, #126352, … |
| maximal normal subgroup of | #126284, #126285, #126352, … |
| maximal quotient of | #146009, #146022, #146041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6476);