Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,11,3,13)(4,17,15,31,19,26,20,23)(6,10,21,24,14,22,16,8)(9,25,27,18,28,32,29,30);(1,3,14,21)(2,8,22,12)(4,15,30,25)(5,6,16,7)(9,23,17,29)(10,13,11,24)(18,32,19,20)(26,27,28,31);(1,4)(2,9,10,26)(3,15)(5,20)(6,18)(7,19)(8,23,13,27)(11,28,22,17)(12,29,24,31)(14,30)(16,32)(21,25)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=d,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eg,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133182, #133183, #133190, … |
| maximal normal subgroup of | #133182, #133183, #133190, … |
| maximal quotient of | #99148, #99151, #99152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1747);