Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 8192 |
| 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 | C2×#102 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
169 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 169 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,18,14,19,28,36,15,20,29,37,27,35,12,17)(11,21,23,33,24,34,22,38,25,39,26,16,30,31,40,32);(2,5)(6,7)(9,11,14,30,15,25,27,24)(10,16,19,38,20,33,35,32)(12,26,13,22,28,23,29,40)(17,34,18,21,36,31,37,39);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=f,b^2=defg,c^2=e,d^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131680, #131681, #131696, … |
| maximal normal subgroup of | #131680, #131681, #131696, … |
| maximal quotient of | #141855, #141856, #141857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6262);