Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | #3475 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(2,5,6,7)(10,11)(12,13)(15,16);(1,2)(3,7)(4,6)(5,8)(9,10,14,16)(11,12,15,13);(9,11)(10,12)(13,16)(14,15)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8438, #186491, #186493, … |
| maximal normal subgroup of | C2×#8438, #186491, #186493, … |
| maximal quotient of | #97487, #97490, #97491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1426);