Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #394 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,20)(13,21)(14,17)(15,19)(16,23)(22,24);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,16)(13,22)(14,20)(17,24)(18,21)(19,23);(2,5)(3,8)(6,7)(9,12,13,15,14,18,23,24)(10,11,16,19,17,20,21,22)Polycyclic presentation:
2^8|c^2=d,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=egh,f^b=fh,e^c=egh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125709, #125710, #125711, … |
| maximal normal subgroup of | #125709, #125710, #125711, … |
| maximal quotient of | #104857, #104858, #104859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3534);