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 | |
4456 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)(11,12)(13,21)(14,18)(15,16)(17,24)(19,22)(20,23)(26,31)(27,34)(28,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,19)(10,12,16,22)(13,23,24,14)(17,18,21,20)(25,26)(27,29)(28,37)(30,31)(32,36)(33,34)(35,38)(39,40);(2,3)(6,7)(10,16)(11,19)(13,24)(18,20)(25,27,30,34)(26,29,31,33)(28,35,36,40)(32,39,37,38);(4,8)(6,7)(9,12)(10,11)(13,20)(14,17)(15,22)(16,19)(18,24)(21,23)(25,28)(26,32)(27,35)(29,39)(30,36)(31,37)(33,38)(34,40);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,14)(12,20)(15,24)(16,21)(18,22)(19,23)(25,27)(26,33)(28,35)(29,31)(30,34)(32,38)(36,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,13)(12,21)(15,23)(16,20)(17,22)(19,24)(25,29,30,33)(26,27,31,34)(28,38,36,39)(32,40,37,35)Polycyclic presentation:
2^9|b^2=gh,c^2=g,e^2=ghi,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, 8519983);