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 | |
2295 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)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(5,6,7,8)(10,12)(15,17)(16,20)(23,24)(28,37)(31,36)(34,40)(38,39);(5,6)(7,8)(9,11,14,19)(10,15,12,17)(13,21,22,18)(16,24,20,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2)(3,4)(9,12,14,10)(11,17,19,15)(13,16,22,20)(18,23,21,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40);(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=f,e^2=f,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, 3242149);