Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32)(34,39)(36,40)(37,45)(42,46);(11,15)(13,16)(18,19)(20,27)(21,29)(30,31)(33,34,38,39)(35,36,43,40)(37,41,45,47)(42,44,46,48);(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,38,43)(34,40,39,36)(37,46,45,42)(41,44,47,48);(1,2)(3,5)(4,6)(7,8)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,40)(34,35,39,43)(37,44,45,48)(41,42,47,46);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,41,39,47)(35,42,43,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=h,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5069109);