Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #383 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2)(3,4)(5,6)(7,15,17,11)(8,12)(9,14)(10,13)(16,20,18,19);(5,7,9,18,10,17,8,16)(6,11,14,20,13,15,12,19);(2,4)(6,12,13,14)(11,19,15,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93545, #134461, #134462, … |
| maximal normal subgroup of | #93545, #134461, #134462, … |
| maximal quotient of | #108529, #108530, #108531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3813);