Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19254 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
319 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,23)(24,31)(25,26)(27,32);(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #271908, #271912, #271924, … |
| maximal normal subgroup of | #271908, #271912, #271924, … |
| maximal quotient of | #189106, #197784, #197787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12265);