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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,7,4)(5,8,14,16)(11,15,13,12)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,3,6,7)(2,4,9,10)(5,13,14,11)(8,15,16,12)(18,24)(19,21)(26,30)(27,29)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,2,6,9)(3,4,7,10)(5,8,14,16)(11,12,13,15)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,13)(11,16)(14,15)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5,6,14)(2,8,9,16)(3,11,7,13)(4,12,10,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297673);