Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10144 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3151 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
307 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,24)(22,23);(1,2,5,8)(3,7,4,6)(10,15)(11,20)(12,13)(17,18)(19,23)(21,22);(2,6)(3,4)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24)Polycyclic presentation:
2^8|b^2=h,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,c^b=cf,d^b=dg,e^b=egh,f^b=fg,d^c=dh,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10148, #236913, #236923, … |
| maximal normal subgroup of | C2×#10148, #236913, #236923, … |
| maximal quotient of | #102591, #102609, #102616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3136);