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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,7)(9,10,13,12)(11,15,16,14)(26,30)(28,35)(31,36)(34,39);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,37)(34,39);(2,6)(3,7)(9,11,13,16)(10,15,12,14)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,14)(13,16)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4)(2,7)(3,6)(5,8)(9,12,13,10)(11,14,16,15)(17,19,20,23)(18,21,22,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,c^2=fi,d^2=gi,e^2=ghi,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, 1773735);