Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| 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 | |
3302 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)(9,10,14,18)(11,16,21,15)(12,13,22,17)(19,23,24,20)(25,26,30,33)(27,28,35,31)(29,32,39,40)(34,36,38,37);(9,11,14,21)(10,15,18,16)(12,19,22,24)(13,20,17,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(6,7)(9,11)(10,16)(12,19)(13,23)(14,21)(15,18)(17,20)(22,24)(25,28,30,31)(26,27,33,35)(29,37,39,36)(32,38,40,34);(1,2,3,4)(5,7,8,6)(9,12,14,22)(10,17,18,13)(11,19,21,24)(15,23,16,20)(25,27)(26,28)(29,38)(30,35)(31,33)(32,37)(34,39)(36,40);(9,13,14,17)(10,12,18,22)(11,20,21,23)(15,19,16,24)(25,29,30,39)(26,32,33,40)(27,34,35,38)(28,36,31,37)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=i,e^2=f,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, 4942447);