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) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #121 |
| 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 | |
185 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 77 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)(5,6,8,11)(7,9,12,10)(14,15)(16,17)(19,20);(1,3,4,2)(13,14,16,15,18,20,17,19);(6,9)(7,12)(10,11)(13,15,16,20,18,19,17,14)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^c=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9575, #202725, #202733, … |
| maximal normal subgroup of | C2×#9575, #202725, #202733, … |
| maximal quotient of | #101909, #101925, #108698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2563);