Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
225 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(2,4)(5,6,9,15,10,14,24,29,11,13,23,28,8,21,22,34)(7,16,18,31,19,26,33,35,20,30,32,36,17,12,25,27);(1,2,3,4)(5,7,8,17,11,20,10,19)(6,12,14,16,13,26,21,30)(9,18,22,25,23,32,24,33)(15,27,29,31,28,35,34,36);(6,13)(9,23)(12,26)(14,21)(16,30)(18,32)(22,24)(25,33)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93327, #93328, #93581, … |
| maximal normal subgroup of | #93327, #93328, #93581, … |
| maximal quotient of | #134222, #134250, #134277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5778);