Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(18,21)(19,23)(25,26,30,31)(27,28,32,35)(29,33,38,39)(34,36,37,40);(10,13)(14,16)(17,18,20,21)(19,22,23,24)(26,31)(28,35)(33,39)(36,40);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,18)(19,24)(20,21)(22,23)(25,27,30,32)(26,28,31,35)(29,37,38,34)(33,40,39,36);(1,2)(3,6)(4,7)(5,8)(25,28)(26,32)(27,31)(29,36)(30,35)(33,37)(34,39)(38,40);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=fhi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2450046);