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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1899 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)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,40,37,36);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,20,22,23)(26,31)(27,34)(33,39)(35,37);(1,3,4,8)(2,5,6,7)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,23,24,20)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,28,30,32)(26,27,31,34)(29,36,38,40)(33,35,39,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,29,30,38)(26,33,31,39)(27,35,34,37)(28,36,32,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,c^2=g,d^2=fg,e^2=fg,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, 4523575);