Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C4×#63011 (order 1024) · Inn(G) = #121 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1923 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,11,27,30,8,13,28,37,5,26,15,41)(3,14,18,39,19,33,48,58,20,40,49,61,17,9,29,32)(4,21,24,52,25,51,53,62,23,10,34,36,22,35,42,55)(16,43,46,63,47,31,56,57,45,38,59,60,44,50,54,64);(1,3,15,29,5,17,28,49,8,20,27,48,7,19,6,18)(2,9,30,39,26,40,12,32,13,33,41,61,11,14,37,58)(4,16,42,54,22,44,34,59,23,45,53,56,25,47,24,46)(10,31,55,60,51,43,36,57,21,50,62,63,35,38,52,64);(1,4,5,22,8,23,7,25)(2,10,11,35,13,21,26,51)(3,16,17,44,20,45,19,47)(6,24,15,42,28,34,27,53)(9,31,14,38,33,50,40,43)(12,36,30,55,37,52,41,62)(18,46,29,54,49,59,48,56)(32,57,39,60,58,64,61,63)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13123, #494833, #495804, … |
| maximal normal subgroup of | C2×#13123, #494833, #495804, … |
| maximal quotient of | #140104, #140105, #140116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6111);