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) = C25×C4 |
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 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,15)(13,16)(17,18,22,20)(19,23,21,24)(25,26,30,31)(27,33,35,28)(29,32,37,39)(34,38,40,36);(3,8)(5,7)(17,19,22,21)(18,23,20,24)(26,31)(28,33)(29,37)(34,40);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,20,22,18)(19,23,21,24)(25,27,30,35)(26,28,31,33)(29,34,37,40)(32,36,39,38);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,21,22,19)(18,23,20,24)(25,28)(26,27)(29,38)(30,33)(31,35)(32,40)(34,39)(36,37);(1,2)(3,5)(4,6)(7,8)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,38)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,d^2=i,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, 3077241);