Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,16,29)(4,18)(5,9)(6,19)(7,11)(8,15,25,32)(10,20)(13,30,24,21)(14,28,17,26)(22,23)(27,31);(1,3)(2,8)(4,13)(5,14)(6,21)(7,16)(9,24)(10,26)(11,25)(12,22)(15,31)(17,18)(19,29)(20,32)(23,30)(27,28);(1,4,18,2,7,9,5,11)(3,13,17,8,16,24,14,25)(6,22,31,27,23,19,20,10)(12,15,28,30,29,32,26,21);(1,5,7,18)(2,9,11,4)(3,14,16,17)(6,20,23,31)(8,24,25,13)(10,22,27,19)(12,28,29,26)(15,21,32,30)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,d^2=h,e^2=g,g^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #496130, #496132, #497426, … |
| maximal normal subgroup of | #496130, #496132, #497426, … |
| maximal quotient of | #483592, #483594, #483596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26459);