Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | Aut(G) order 32768 |
| 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 | |
283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,17,27,32)(11,19,21,33,22,15,26,29)(12,23,25,37,24,16,30,31)(20,28,36,39,35,34,38,40);(9,11,14,22)(10,15,17,19)(12,20,24,35)(13,26,27,21)(16,28,23,34)(18,33,32,29)(25,38,30,36)(31,40,37,39);(1,3,4,8)(2,5,6,7)(9,12)(10,16,17,23)(11,20)(13,25)(14,24)(15,28,19,34)(18,31,32,37)(21,36)(22,35)(26,38)(27,30)(29,39,33,40)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=dh,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 | #94559, #133443, #133444, … |
| maximal normal subgroup of | #94559, #133443, #133444, … |
| maximal quotient of | #99147, #99150, #99156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1742);