Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,5)(3,8,4,7)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,25,27,21)(20,28,31,24)(26,30,29,32)(36,43)(39,46)(41,47)(44,48);(18,23)(21,25)(24,28)(30,32)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,3,6,4)(2,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,29)(22,27)(23,25)(24,32)(26,31)(28,30)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4,6,3)(2,8,5,7)(17,20,22,31)(18,24,23,28)(19,26,27,29)(21,30,25,32)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,13,14,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,29)(31,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fi,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4264778);