Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2092 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,4,9,14,10,30,28,6,12,22,27,5,11,21,8)(3,13,16,31,18,23,32,26,19,24,17,25,15,29,20,7);(1,3,14,15,6,19,5,18)(2,7,10,25,12,26,11,31)(4,20,30,17,22,32,21,16)(8,13,9,29,28,24,27,23);(2,8)(3,15)(4,21)(5,14)(7,23)(9,11)(10,27)(12,28)(13,31)(16,17)(18,19)(22,30)(24,25)(26,29)Polycyclic presentation:
2^8|a^2=d,b^2=de,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dg,e^b=egh,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13523, #499518, #499603, … |
| maximal normal subgroup of | C2×#13523, #499518, #499603, … |
| maximal quotient of | #146186, #146188, #146200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6511);