Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) = C29×D4 (order 4096) · Inn(G) = C2×D4 · Out(G) = C28 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
184 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,15,17,19,16,20);(6,9)(7,12)(10,11)(13,15,18,19,17,20,16,14);(1,2,3,4)(6,10)(9,11)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eg,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9606, #203625, #203626, … |
| maximal normal subgroup of | C2×#9606, #203625, #203626, … |
| maximal quotient of | #101941, #101944, #101947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2594);