Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(5,6,7,8)(10,14)(12,20)(15,17)(18,24)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,38,40,36);(6,8)(9,10)(11,17)(12,21)(13,14)(15,19)(16,20)(18,22)(23,24)(25,28,31,32)(26,27,34,29)(30,38,39,36)(33,37,40,35);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,40,38);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928627);