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 | |
1433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,22,24,23)(26,30)(27,33)(28,35)(38,40);(9,11)(10,14)(12,15)(13,16)(17,18,21,20)(19,23,24,22)(26,30)(27,33)(32,37)(34,39);(9,11,13,16)(10,15,12,14)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,14)(13,16)(17,20,21,18)(19,23,24,22)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,14,16,15)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fgh,c^2=gh,d^2=fghi,e^2=fgi,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, 1838580);