Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | Aut(G) order 32768 |
| 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 | |
468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 468 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:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,2,6,10)(3,8,13,16)(4,9,5,7)(11,15,12,14)(17,19,22,29)(18,23,26,31)(20,28,21,30)(24,27,25,32);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,15)(10,16)(20,21)(24,25)(27,32)(28,30);(1,4,6,5)(2,9,10,7)(3,12,13,11)(8,14,16,15)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153502, #249742, #249748, … |
| maximal normal subgroup of | #153502, #249742, #249748, … |
| maximal quotient of | #168124, #168129, #168151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12068);