Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
143 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)(3,6)(4,5)(7,8)(9,10)(11,21,25,36)(12,27,26,40)(13,29)(14,30)(15,20)(16,24,34,37)(17,28,35,38)(18,23)(19,31)(22,33)(32,39);(2,5)(6,8)(9,11,22,37,14,26,39,28,15,25,13,24,31,12,23,38)(10,16,32,36,19,17,33,40,20,34,18,21,30,35,29,27);(1,3,4,7)(2,6,5,8)(9,12,13,28,14,11,23,24,15,26,22,38,31,25,39,37)(10,17,18,27,19,34,29,36,20,35,32,40,30,16,33,21)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^2=d,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #134989, #134993, #135007, … |
| maximal normal subgroup of | #134989, #134993, #135007, … |
| maximal quotient of | #135649, #135650, #135671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5875);