Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
205 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 205 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,7,6,10,4,9)(3,11,13,8,14,15,12,16)(17,18,22,26,23,25,21,19)(20,27,28,24,29,32,30,31);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,22,25,23,26,21,18)(20,27,28,31,29,32,30,24);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20,23,29)(18,24,25,31)(19,27,26,32)(21,30,22,28)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9288, #198932, #198936, … |
| maximal normal subgroup of | C2×#9288, #198932, #198936, … |
| maximal quotient of | #99933, #99934, #99949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2276);