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 | #58800 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(18,23)(24,25)(26,29)(31,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,8)(3,4)(10,15)(11,12)(17,18,22,23)(19,24,20,25)(21,26,30,29)(27,31,28,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,20,22,19)(18,25,23,24)(21,27,30,28)(26,31,29,32)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8602510);