Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | Aut(G) order 16384 |
| 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 | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
292 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(17,18,20,25,22,26,21,23)(19,24,27,31,29,32,28,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,20,22,21)(18,25,26,23)(19,28,29,27)(24,30,32,31)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^c=dg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139196, #139200, #300400, … |
| maximal normal subgroup of | #139196, #139200, #300400, … |
| maximal quotient of | #189595, #197847, #202830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12347);