Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10631 |
| automorphism | Aut(G) order 8192 |
| 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 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
155 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(9,10,14,19,15,20,13,18)(11,21,24,16,25,32,23,33)(12,26,28,34,29,35,30,17)(22,37,38,40,27,31,39,36);(1,2)(3,5)(4,7)(6,8)(9,11,15,25)(10,16,20,33)(12,27,29,22)(13,24,14,23)(17,31,34,37)(18,32,19,21)(26,40,35,36)(28,38,30,39);(2,5)(3,8)(6,7)(9,12,13,28,15,29,14,30)(10,17,18,26,20,34,19,35)(11,22,23,38,25,27,24,39)(16,31,21,36,33,37,32,40)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=d,d^2=g,e^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,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126296, #126350, #126388, … |
| maximal normal subgroup of | #126296, #126350, #126388, … |
| maximal quotient of | #105211, #105212, #105216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3620);