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) = C23 |
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 | |
199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,6,12,13)(3,9,11,16,10,7,14,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(2,6)(4,12)(7,9)(11,14)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3)(2,7,6,9)(4,11)(5,10)(8,15,13,16)(12,14)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=d,f^2=g,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 | #132131, #132132, #132133, … |
| maximal normal subgroup of | #132131, #132132, #132133, … |
| maximal quotient of | #99148, #99153, #99158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1743);