Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12304 |
| automorphism | Aut(G) = #72308439 (order 2048) · Inn(G) = #1616 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3259 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,11,6,12,5,22)(3,13,23,30,15,9,16,26)(4,17,18,27,20,8,21,10)(14,24,31,25,19,28,32,29);(2,8)(4,18)(5,7)(9,24)(10,22)(11,27)(12,17)(13,28)(14,31)(16,23)(19,32)(20,21)(25,30)(26,29);(1,3,6,15)(2,9)(4,19,20,14)(5,16,7,23)(8,24)(10,25)(11,26)(12,13)(17,28)(18,32,21,31)(22,30)(27,29)Polycyclic presentation:
2^8|a^2=fg,c^2=g,d^2=f,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12304, #484982, #484983, … |
| maximal normal subgroup of | C2×#12304, #484982, #484983, … |
| maximal quotient of | #126351, #126389, #126448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5292);