Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| automorphism | Aut(G) order 8192 |
| 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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(1,2,6,9)(3,10,8,5)(4,7,12,14)(11,16,15,13)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,2,6,8,10,9)(4,11,13,7,12,15,16,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,4)(2,7)(3,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(4,12)(7,14)(11,15)(13,16)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19521, #6354612, #6354641, … |
| maximal normal subgroup of | C2×#19521, #6354612, #6354641, … |
| maximal quotient of | #189611, #197921, #198006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12509);