Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
235 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 235 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) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,10)(7,8)(11,12)(14,17)(15,19)(18,20);(1,3,4,2)(5,6,9,11)(7,10,8,12)(15,19)(17,20);(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,18)(17,19)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8887, #192594, #192613, … |
| maximal normal subgroup of | C2×#8887, #192594, #192613, … |
| maximal quotient of | #99360, #99377, #99387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1875);