Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3109 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
470 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,29)(28,32)(30,31);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,22,30)(18,24,25,32)(20,29,21,28)(23,31,26,27);(2,9)(7,16)(8,13)(10,15)(18,25)(23,26)(24,32)(27,31)Polycyclic presentation:
2^8|b^2=gh,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #246948, #246952, #246978, … |
| maximal normal subgroup of | #246948, #246952, #246978, … |
| maximal quotient of | #165885, #165929, #165952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11524);