Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(6,7)(9,10,11,14,12,13,15,16);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(10,13)(11,15)Polycyclic presentation:
2^8|a^2=g,c^2=d,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97225, #97226, #97228, … |
| maximal normal subgroup of | #97225, #97226, #97228, … |
| maximal quotient of | #118694, #118695, #119452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4186);