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 | #3253 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
301 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 301 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)(3,10,13,16)(4,8,5,14)(7,11,15,12)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,d^2=g,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 | #331937, #331941, #331945, … |
| maximal normal subgroup of | #331937, #331941, #331945, … |
| maximal quotient of | #189616, #197914, #197996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12507);