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, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,20,24,21,23,27,30)(19,22,25,29,26,28,31,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,21,26)(18,22,23,28)(20,25,27,31)(24,29,30,32);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(18,23)(20,27)(22,28)(25,31)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #133184, #133185, #133192, … |
| maximal normal subgroup of | #133184, #133185, #133192, … |
| maximal quotient of | #99148, #99149, #99150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1746);