Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
197 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,13)(14,15)(17,18,22,23,21,26,20,25)(19,27,30,24,29,32,28,31);(1,3,4,8)(2,5,6,7)(9,10,11,13)(12,14,16,15)(18,23)(20,22)(24,27)(25,26)(28,30)(31,32);(10,14)(12,16)(13,15)(17,19,21,29)(18,24,26,31)(20,30,22,28)(23,27,25,32)Polycyclic presentation:
2^8|a^2=dgh,b^2=deg,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10682, #243948, #243955, … |
| maximal normal subgroup of | C2×#10682, #243948, #243955, … |
| maximal quotient of | #105240, #117098, #117099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3670);