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 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,21)(12,14)(13,19)(15,17)(16,23)(18,20)(22,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2)(3,4)(5,6)(7,8)(9,11,15,23)(10,16,17,21)(12,22,20,13)(14,19,18,24)(26,27)(31,33)(32,34)(38,39);(2,6)(3,7)(10,17)(12,20)(13,22)(16,21)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(3,7)(4,8)(9,12)(10,18)(11,22)(13,23)(14,17)(15,20)(16,24)(19,21)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,22)(10,19,17,24)(11,20,23,12)(14,21,18,16);(1,4)(2,3)(5,8)(6,7)(9,14,15,18)(10,20,17,12)(11,19,23,24)(13,21,22,16)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532643);