Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 32768 |
| 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 | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 291 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)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,22)(19,26)(20,23)(25,30)(27,28)(29,32);(17,18,23,30,21,25,20,22)(19,24,26,32,27,31,28,29);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,21,27)(18,24,25,31)(20,28,23,26)(22,29,30,32);(19,27)(24,31)(26,28)(29,32)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138597, #138604, #143348, … |
| maximal normal subgroup of | #138597, #138604, #143348, … |
| maximal quotient of | #189569, #197767, #202766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12197);