Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| 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 | #3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
466 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 466 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(20,29)(23,28)(25,32)(27,30);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^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,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249811, #249821, #249841, … |
| maximal normal subgroup of | #249811, #249821, #249841, … |
| maximal quotient of | #168125, #168151, #168216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11712);