Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
175 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,14,12,16,11,15);(1,3,5,4)(2,6,7,8)(9,11,12,13)(10,14,16,15)(17,18)(19,21)(20,23)(22,24);(1,4,5,3)(2,6,7,8)(10,15)(11,13)(14,16)(18,21)(19,24)(22,23)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=dg,e^2=g,f^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fh,e^c=eg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #126325, #126326, #126327, … |
| maximal normal subgroup of | #126325, #126326, #126327, … |
| maximal quotient of | #120139, #120254, #120749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3638);