Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17,12,19,30,18)(7,20,24,33,8,25,28,36)(9,29,31,15,21,16,10,32)(13,27,14,26,35,23,34,22);(5,7,21,26,30,28,31,27,12,8,9,22,11,24,10,23)(6,13,15,20,17,14,16,33,19,35,32,25,18,34,29,36);(2,4)(5,8,9,26,30,24,10,27,12,7,21,22,11,28,31,23)(6,14,15,33,17,35,16,25,19,34,32,36,18,13,29,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13152, #494919, #494920, … |
| maximal normal subgroup of | C2×#13152, #494919, #494920, … |
| maximal quotient of | #140176, #140177, #140199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6140);