Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C2×#72231746 (order 2048) · Inn(G) = #99 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | C4×#99 |
| prime radical | #525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,4,9,18)(3,12,15,29,16,26,23,32)(5,10,20,27,21,17,25,31)(8,22,24,19,13,28,30,14);(1,3)(2,8)(4,13)(5,19,21,14)(6,15)(7,16)(9,23)(10,26,17,12)(11,24)(18,30)(20,28,25,22)(27,32,31,29);(1,4,6,18,7,2,9,11)(3,13,15,30,16,8,23,24)(5,17,20,31,21,10,25,27)(12,28,29,14,26,22,32,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=h,b^a=bd,f^a=fg,d^b=dg,g^b=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94381, #94382, #128461, … |
| maximal normal subgroup of | #94381, #94382, #128461, … |
| maximal quotient of | #128308, #128344, #128366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5611);