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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
680 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)(10,14)(11,17)(16,21)(19,23)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,3,5,8)(2,6,7,4)(12,20)(16,21)(19,23)(22,24)(27,35)(29,39)(31,40)(33,36);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(1,3)(2,4)(5,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,d^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×#54237, #8101595, #8101965, … |
| maximal normal subgroup of | C2×#54237, #8101595, #8101965, … |
| maximal quotient of | #585842, #585886, #587192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47225);