Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #381 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(5,6,9,15,10,13,8,14)(7,16,17,20,18,11,19,12);(1,2)(3,4)(5,7)(6,11)(8,17)(9,19)(10,18)(12,14)(13,16)(15,20);(2,4)(5,7)(6,12,13,20)(8,17)(9,19)(10,18)(11,14,16,15)Polycyclic presentation:
2^8|a^2=g,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133478, #133479, #133480, … |
| maximal normal subgroup of | #133478, #133479, #133480, … |
| maximal quotient of | #118665, #118699, #118702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4137);