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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
269 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)(3,11,13,7)(4,15,17,21)(5,9)(6,10)(8,16,22,32)(12,26,28,24)(14,25,30,20)(18,23)(19,27,31,29);(1,3,6,14)(2,7,10,20)(4,12,17,29)(5,13,18,30)(8,19,22,26)(9,11,23,25)(15,31,21,24)(16,27,32,28);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,24)(10,22,23,15)(13,27)(14,29)(18,32)(20,26,25,31)(28,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=d,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 | #132139, #132140, #132141, … |
| maximal normal subgroup of | #132139, #132140, #132141, … |
| maximal quotient of | #99148, #99153, #99158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1738);