Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(18,22)(19,25)(20,27)(30,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(2,5)(6,8)(18,22)(19,25)(24,29)(26,31)(33,34,37,40)(35,38,42,47)(36,39,45,43)(41,46,44,48);(1,2,4,5)(3,6,7,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,46)(42,47)(43,45)(44,48);(9,11,12,16)(10,13,14,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,3)(2,6)(4,7)(5,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fgh,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802556);