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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,12)(5,8,14,16)(10,11,13,15)(17,18)(19,21)(20,28)(22,30)(23,24)(25,29)(26,31)(27,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,6,7)(2,4,9,12)(5,13,14,10)(8,15,16,11)(18,24)(21,25)(22,32)(26,28)(34,39)(35,40)(41,46)(42,47);(1,3)(2,4)(5,10)(6,7)(8,11)(9,12)(13,14)(15,16)(17,19,23,29)(18,25,24,21)(20,30,31,27)(22,28,32,26)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(17,20,23,31)(18,26,24,28)(19,27,29,30)(21,22,25,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,13)(10,16)(14,15)(17,21,23,25)(18,19,24,29)(20,22,31,32)(26,27,28,30);(1,5)(2,8)(3,10)(4,11)(6,14)(7,13)(9,16)(12,15)(17,22)(18,27)(19,28)(20,25)(21,31)(23,32)(24,30)(26,29)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9152321);