Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(26,30)(27,33)(28,36)(38,40);(1,2)(3,4)(5,6)(7,8)(12,20)(15,22)(18,24)(21,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(2,6)(3,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3,5,7)(2,4,6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,4)(2,7)(3,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=fg,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58116, #8104582, #8104698, … |
| maximal normal subgroup of | C2×#58116, #8104582, #8104698, … |
| maximal quotient of | #585681, #586115, #586646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51104);