Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1400 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)(3,4)(5,8)(6,7)(9,10,13,11)(12,15,16,14)(18,22)(23,25)(24,29)(30,31)(36,43)(39,46)(41,47)(44,48);(2,7)(3,5)(20,28)(24,29)(26,32)(30,31)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,3,6,5)(2,8,7,4)(9,11,13,10)(12,15,16,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,3)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,5,6,3)(2,8,7,4)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5791609);