Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| 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 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,5)(4,7)(11,15)(14,16)(19,23)(21,24)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(1,2)(3,4)(5,7)(6,8)(10,13)(14,16)(25,27,30,32)(26,31,34,28)(29,35,38,37)(33,39,40,36);(1,3,6,5)(2,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,27)(26,31)(28,34)(29,37)(30,32)(33,36)(35,38)(39,40);(1,4,6,7)(2,3,8,5)(17,19)(18,21)(20,23)(22,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,5,6,3)(2,7,8,4)(9,11,12,15)(10,14,13,16)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4931816);