Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,21,23,24)(25,26)(27,32)(28,36)(29,37)(30,31)(33,34)(35,39)(38,40);(9,10,13,11)(12,15,16,14)(17,19,20,23)(18,21,22,24)(26,31)(29,36)(32,39)(34,38);(10,11)(14,15)(17,19)(18,21)(20,23)(22,24)(25,27,30,35)(26,32,31,39)(28,33,37,40)(29,38,36,34);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(25,28)(26,29)(27,33)(30,37)(31,36)(32,38)(34,39)(35,40);(1,3,4,7)(2,5,6,8)(9,12,13,16)(10,14,11,15)(17,18)(19,21)(20,22)(23,24)(25,29)(26,28)(27,34)(30,36)(31,37)(32,40)(33,39)(35,38)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=f,d^2=g,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5484024);