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 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
220 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 220 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:
(1,2)(3,4)(6,11)(7,12)(8,9)(13,14,17,15,16,18,20,19);(2,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,18,16,19,20,14);(5,7,10,11)(6,8,12,9)(14,18)(15,19)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,e^b=eg,d^c=dg,e^c=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8848, #191877, #191879, … |
| maximal normal subgroup of | C2×#8848, #191877, #191879, … |
| maximal quotient of | #99349, #99353, #99354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1836);