Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| automorphism | Aut(G) order 16777216 |
| 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 | |
3573 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)(6,8)(11,17)(12,19)(14,20)(15,22)(26,31)(28,33)(29,37)(34,40);(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,31)(27,33,35,28)(29,32,37,38)(34,39,40,36);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27,30,35)(26,28,31,33)(29,34,37,40)(32,36,38,39);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,14,16,20)(12,18,19,23)(15,21,22,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4216275);