Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1902 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,22,23,24)(27,33)(28,36)(30,34)(31,38);(2,4)(3,8)(11,16)(13,15)(18,21)(19,23)(25,26,29,32)(27,34,33,30)(28,31,36,38)(35,40,39,37);(3,8)(6,7)(17,18,20,21)(19,22,23,24)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5899424);