Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10637 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #397 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
149 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,7,8,5)(9,10)(11,19,22,32)(12,16)(13,17)(14,18)(15,21,29,26)(20,35,23,34)(24,30)(25,31)(27,33)(28,37,39,36)(38,40);(9,11,14,22)(10,15,18,29)(12,23,25,20)(13,26,27,21)(16,28,31,39)(17,32,33,19)(24,36,38,37)(30,35,40,34);(2,5)(3,8)(6,7)(9,12,13,24,14,25,27,38)(10,16,17,30,18,31,33,40)(11,20,21,36,22,23,26,37)(15,28,19,34,29,39,32,35)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,e^c=eh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10639, #243014, #243024, … |
| maximal normal subgroup of | C2×#10639, #243014, #243024, … |
| maximal quotient of | #105211, #105213, #105221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3627);