Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(12,19)(16,21)(18,23)(22,24)(26,28)(27,33)(32,36)(35,39);(10,14)(11,17)(16,21)(18,23)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37062, #7634338, #7634391, … |
| maximal normal subgroup of | C2×#37062, #7634338, #7634391, … |
| maximal quotient of | #553947, #553969, #554155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30050);