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 | C2×#127 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,22)(19,26,21,24)(20,28,29,25)(27,31,30,32)(34,38)(35,41)(40,43)(42,44);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(2,6)(7,14)(8,11)(15,16)(20,29)(25,28)(27,30)(31,32)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,19)(18,24,22,26)(20,30,29,27)(25,31,28,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22,23,18)(19,26,21,24)(20,25,29,28)(27,31,30,32)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633403);