Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
251 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 127 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)(5,6)(7,8)(9,11)(10,12)(13,14,16,19,17,15,18,20);(2,4)(5,7,9,12,10,11,8,6);(5,8,10,9)(6,7,12,11)(13,15,16,20,17,14,18,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fg,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8201, #182401, C8×#419, … |
| maximal normal subgroup of | C2×#8201, #182401, C8×#419, … |
| maximal quotient of | #96687, #96693, #96710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1189);