Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2264 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)(9,10,13,14)(11,17,19,15)(12,16,21,22)(18,24,20,23)(25,26,30,33)(27,34,37,31)(28,29,39,32)(35,36,38,40);(10,14)(15,17)(16,22)(23,24)(25,27,30,37)(26,31,33,34)(28,38,39,35)(29,36,32,40);(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,27)(26,31)(28,35)(29,40)(30,37)(32,36)(33,34)(38,39);(1,2,3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,28)(26,32)(27,35)(29,33)(30,39)(31,36)(34,40)(37,38);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29,30,32)(26,28,33,39)(27,36,37,40)(31,38,34,35)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296667);