Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
3484 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)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(26,31)(28,32)(34,39)(36,37);(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327729);