Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| 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×#102 |
| 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 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,21,16,22,15,20)(11,23,28,40,29,39,27,35)(12,30,24,18,31,19,13,32)(17,34,36,26,38,25,37,33);(2,5)(6,7)(9,11,24,26,15,28,12,25,16,29,13,33,14,27,31,34)(10,17,19,39,21,37,32,40,22,38,30,23,20,36,18,35);(1,3,4,8)(2,6,5,7)(9,12,14,24,16,31,15,13)(10,18,20,30,22,32,21,19)(11,25,27,26,29,34,28,33)(17,35,36,23,38,40,37,39)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131922, #131929, #131930, … |
| maximal normal subgroup of | #131922, #131929, #131930, … |
| maximal quotient of | #141887, #141890, #141915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6277);