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, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(10,14)(12,19)(15,20)(18,23)(25,26,31,34)(27,29,28,32)(30,38,39,33)(35,40,36,37);(10,14)(11,17)(16,21)(18,23)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fgh,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×#37090, #7634359, #7634388, … |
| maximal normal subgroup of | C2×#37090, #7634359, #7634388, … |
| maximal quotient of | #553955, #553973, #554157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30078);