Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,35)(27,33,28,32)(29,30,34,40)(36,39,38,37);(11,17)(14,21)(18,23)(22,24)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,27)(26,33,35,32)(29,36,34,38)(30,37,40,39);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,29)(26,30)(27,36)(28,38)(31,34)(32,37)(33,39)(35,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4932948);