Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
295 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(1,2,6,4)(3,9,10,16)(5,13,15,8)(7,14,12,11)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3,6,10)(2,7,4,12)(5,14,15,11)(8,16,13,9)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(3,10)(7,12)(9,16)(11,14)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,4,6,2)(3,9,10,16)(5,8,15,13)(7,14,12,11)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #492101, #492103, #492117, … |
| maximal normal subgroup of | #492101, #492103, #492117, … |
| maximal quotient of | #220799, #220971, #220983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12954);