Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C2×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
270 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,5,7,4,3,6,8)(9,10,13,16,14,11,12,15);(1,3,5,8,4,2,6,7)(9,11,14,10)(12,15,13,16);(2,3)(7,8)(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,d^b=dg,f^b=fg,e^c=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8956, #193732, #193733, … |
| maximal normal subgroup of | C2×#8956, #193732, #193733, … |
| maximal quotient of | #99519, #99520, #99521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1944);