Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2125 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)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,28)(27,33,29,32)(30,36,38,34)(35,39,37,40);(6,8)(11,18)(14,17)(19,24)(22,23)(25,27,31,29)(26,32,28,33)(30,37,38,35)(34,40,36,39);(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,2,3,4)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3043500);