Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C4×#60399 (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C23 | C4×C8 |
| Frattini, cosocle | C2×#131 | C22 |
| socle | C2 | #1282 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 2 generators on 64 points:
(1,2,5,11,7,14,27,38,8,13,6,12,26,37,28,39)(3,15,17,40,19,43,48,62,20,42,18,41,47,61,49,63)(4,21,22,36,23,50,51,60,24,10,25,33,52,34,53,58)(9,29,30,54,32,55,56,16,31,44,35,45,57,64,59,46);(1,3,6,18,7,19,28,49,8,20,5,17,26,47,27,48)(2,9,12,30,14,32,39,56,13,31,11,35,37,57,38,59)(4,16,22,44,23,45,51,64,24,46,25,29,52,54,53,55)(10,15,33,41,34,43,58,63,21,42,36,40,50,61,60,62)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,e^2=g,g^2=h,b^a=bc,c^a=cf,f^a=fh,d^b=df,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7207, #127082, #127133, … |
| maximal normal subgroup of | C2×#7207, #127082, #127133, … |
| maximal quotient of | #92995, #92996, #93001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 195);