Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,8)(10,13)(14,16)(21,33)(26,36)(28,42)(31,44)(38,45)(40,46)(41,43)(47,48);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(18,23)(24,34)(25,35)(26,36)(37,39)(38,45)(40,46)(47,48);(1,3)(2,7)(4,6)(5,8)(17,18)(19,24)(20,25)(21,26)(22,23)(27,37)(28,38)(29,34)(30,39)(31,40)(32,35)(33,36)(41,47)(42,45)(43,48)(44,46);(17,19,22,29)(18,24,23,34)(20,30,32,27)(21,28,33,42)(25,39,35,37)(26,38,36,45)(31,43,44,41)(40,48,46,47);(17,20,22,32)(18,25,23,35)(19,27,29,30)(21,31,33,44)(24,37,34,39)(26,40,36,46)(28,41,42,43)(38,47,45,48);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,28)(20,31)(22,33)(23,36)(24,38)(25,40)(27,41)(29,42)(30,43)(32,44)(34,45)(35,46)(37,47)(39,48)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538040);