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 | |
4220 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19,23,27)(18,20,24,30)(21,31,32,22)(25,26,29,28)(34,37)(40,43)(41,45)(47,48);(2,5)(8,10)(9,12)(15,16)(18,24)(19,27)(21,32)(26,28)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,20)(18,19)(21,28)(22,25)(23,30)(24,27)(26,32)(29,31)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,21)(18,25)(19,22)(20,28)(23,32)(24,29)(26,30)(27,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,22)(18,26)(19,21)(20,29)(23,31)(24,28)(25,30)(27,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8509693);