Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
315 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,4,12,6,11,5,13)(2,7,14,16,9,15,8,10);(3,11)(7,15)(10,16)(12,13)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|b^2=eg,c^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #138470, #138479, #143122, … |
| maximal normal subgroup of | #138470, #138479, #143122, … |
| maximal quotient of | #189115, #197796, #202799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12277);