Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 687 conjugacy classes of subgroups — too many to draw. Only its 179 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)(7,14,16,20)(8,12)(9,18)(10,13)(11,15,19,17);(5,7,9,17,10,16,8,15)(6,11,12,14,13,19,18,20);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,g^a=gh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93526, #93527, #93537, … |
| maximal normal subgroup of | #93526, #93527, #93537, … |
| maximal quotient of | #105486, #105487, #105488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3705);