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 | |
721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,8)(10,14)(11,17)(16,21)(18,23)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,2,5,6)(3,8,7,4)(10,14)(15,20)(16,21)(22,24)(26,28)(31,33)(32,36)(38,39);(2,6)(3,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4)(2,7)(3,6)(5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,c^2=fgh,d^2=f,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×#57837, #8101754, #8101966, … |
| maximal normal subgroup of | C2×#57837, #8101754, #8101966, … |
| maximal quotient of | #585246, #587256, #587294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50825);