Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #399 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,7,5,8,4,6)(9,10,13,18,14,19,15,29)(11,20,22,32,24,36,23,16)(12,25,27,35,28,17,30,34)(21,37,26,40,38,33,39,31);(2,6)(3,4)(7,8)(9,11,15,22,14,24,13,23)(10,16,29,20,19,32,18,36)(12,26,30,38,28,39,27,21)(17,33,35,31,25,37,34,40);(1,2,5,8)(3,6,4,7)(9,12,14,28)(10,17,19,25)(11,21,24,38)(13,27,15,30)(16,31,32,40)(18,34,29,35)(20,37,36,33)(22,26,23,39)Polycyclic presentation:
2^8|a^2=d,b^2=de,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #126515, C2×#10668, #243890, … |
| maximal normal subgroup of | #126515, C2×#10668, #243890, … |
| maximal quotient of | #105232, #105237, #105238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3656);