Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #498 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #498 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2058 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,14,29)(4,17)(5,10)(6,22)(7,11)(8,15,24,31)(9,21)(13,26,16,28)(18,32)(19,23,30,27)(20,25);(3,13,14,16)(4,18)(8,23,24,27)(9,25)(12,28,29,26)(15,19,31,30)(17,32)(20,21);(1,3,6,13,7,14,5,16)(2,8,10,27,11,24,22,23)(4,19,21,15,18,30,20,31)(9,26,17,12,25,28,32,29)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=dg,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^b=eh,d^c=dh,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125945, #125946, #126214, … |
| maximal normal subgroup of | #125945, #125946, #126214, … |
| maximal quotient of | #145796, #145797, #145826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6447);