Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = 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 | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20,13,18,29,21)(11,19,24,33,23,16,14,30)(12,25,27,37,26,40,28,32)(17,34,36,39,35,31,38,22);(2,5)(6,8)(9,11,15,24,13,23,29,14)(10,16,20,30,18,19,21,33)(12,22,27,34,26,39,28,31)(17,32,36,25,35,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,13,26)(10,17,18,35)(11,22,23,39)(14,31,24,34)(15,27,29,28)(16,32,19,37)(20,36,21,38)(25,33,40,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^c=eg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13113, #494795, #494796, … |
| maximal normal subgroup of | C2×#13113, #494795, #494796, … |
| maximal quotient of | #140068, #140069, #140070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6101);