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 | |
4518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,14)(13,18)(15,17)(19,20)(21,24)(22,23)(25,26)(27,38)(28,36)(29,31)(30,34)(32,37)(33,40)(35,39);(1,2,5,6)(3,7,8,4)(9,11,15,18)(10,16,17,13)(12,19,21,23)(14,20,24,22)(25,27,32,40)(26,33,37,38)(28,39,30,29)(31,34,35,36);(2,6)(4,7)(10,17)(11,18)(14,24)(19,23)(25,28,32,30)(26,34,37,36)(27,39,40,29)(31,33,35,38);(2,6)(3,8)(9,12)(10,14)(11,19)(13,22)(15,21)(16,20)(17,24)(18,23)(25,29)(26,35)(27,28)(30,40)(31,37)(32,39)(33,36)(34,38);(1,3)(2,4)(5,8)(6,7)(9,13,15,16)(10,11,17,18)(12,20,21,22)(14,23,24,19)(25,30,32,28)(26,36,37,34)(27,39,40,29)(31,33,35,38);(1,4)(2,3)(5,7)(6,8)(9,14)(10,12)(11,20)(13,23)(15,24)(16,19)(17,21)(18,22)(25,31,32,35)(26,29,37,39)(27,34,40,36)(28,38,30,33)Polycyclic presentation:
2^9|b^2=gi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295369);