Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
323 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,9)(7,10)(11,12)(13,14,15,19,17,20,16,18);(5,6,10,12,8,11,7,9)(14,18)(15,16)(19,20);(2,4)(6,11)(9,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9229, #197713, #197721, … |
| maximal normal subgroup of | C2×#9229, #197713, #197721, … |
| maximal quotient of | #99865, #99881, #100343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2217);