Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | Aut(G) = C24×#3484 (order 2048) · Inn(G) = #124 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #400 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,5)(3,6,8,7)(9,10,13,12,14,16,18,22)(11,17,19,24,20,21,23,15);(1,3,4,8)(2,6,5,7)(9,11,18,19,14,20,13,23)(10,15,22,17,16,24,12,21);(2,5)(3,7)(6,8)(9,12,13,16,14,22,18,10)(11,17,19,24,20,21,23,15)Polycyclic presentation:
2^8|a^2=dh,b^2=de,c^2=d,d^2=g,f^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10689, #244080, #244082, … |
| maximal normal subgroup of | C2×#10689, #244080, #244082, … |
| maximal quotient of | #120266, #120643, #120770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3677);