Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1487 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)(3,8)(11,15)(14,16)(17,18)(19,24)(20,23)(21,22)(25,26,30,28)(27,33,34,31)(29,32,38,36)(35,40,37,39);(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(17,19,22,20)(18,23,21,24)(26,28)(27,34)(32,36)(35,37);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,21,24)(25,27,30,34)(26,31,28,33)(29,37,38,35)(32,40,36,39);(1,3,5,8)(2,7,6,4)(25,28,30,26)(27,33,34,31)(29,36,38,32)(35,40,37,39);(1,4,5,7)(2,8,6,3)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,24,20,23)(25,29,30,38)(26,32,28,36)(27,35,34,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=fi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4621394);