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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26,31,33)(27,34,28,29)(30,38,39,32)(35,37,36,40);(1,3)(2,6)(4,7)(5,8)(10,13)(11,17)(12,20)(23,24)(25,27,31,28)(26,29,33,34)(30,35,39,36)(32,37,38,40);(1,4,5,6)(2,3,7,8)(9,11,14,17)(10,15,13,19)(12,21,20,18)(16,24,22,23)(25,28,31,27)(26,29,33,34)(30,36,39,35)(32,37,38,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,20)(10,16,13,22)(11,18,17,21)(15,23,19,24)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)(25,30,31,39)(26,32,33,38)(27,35,28,36)(29,37,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=f,d^2=h,e^2=fg,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×#58462, #8104246, #8104286, … |
| maximal normal subgroup of | C2×#58462, #8104246, #8104286, … |
| maximal quotient of | #585598, #586484, #586620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51450);