Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,5)(3,8,6,4)(9,10,13,12)(11,14,16,15)(19,23)(21,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,3,7,6)(2,8,5,4)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,21,23,24)(26,30)(27,34)(28,37)(38,39);(1,4)(2,6)(3,5)(7,8)(9,11,13,16)(10,14,12,15)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,5,7,2)(3,8,6,4)(17,19,20,23)(18,21,22,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,6,7,3)(2,8,5,4)(9,12,13,10)(11,15,16,14)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=gh,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1590622);