Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2302 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)(10,14)(11,17)(12,19)(22,24)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,36,40,33);(6,8)(11,17)(15,20)(18,23)(22,24)(25,27,30,37)(26,31,34,35)(28,36,39,33)(29,32,40,38);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,31)(28,36)(29,38)(30,37)(32,40)(33,39)(34,35);(1,2,3,4)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,39)(26,32,34,38)(27,36,37,33)(29,31,40,35);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,40)(26,33,34,36)(27,32,37,38)(28,35,39,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3371260);