Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3537 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7,5,8,11,16)(3,6,10,14,9,13,12,15);(2,6)(4,11)(7,14)(8,13)(10,12)(15,16);(3,9)(10,12)Polycyclic presentation:
2^8|a^2=f,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=egh,f^b=fg,d^c=dg,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12755, #493569, #493570, … |
| maximal normal subgroup of | C2×#12755, #493569, #493570, … |
| maximal quotient of | #129599, #129604, #129634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5743);