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 | |
4517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,8)(3,5,6,4)(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,3,7,6)(2,4,8,5)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(27,34)(31,33)(35,40)(38,39);(1,4,7,5)(2,3,8,6)(10,17)(14,21)(16,18)(20,24)(25,27,30,34)(26,31,29,33)(28,35,37,40)(32,38,36,39);(1,5,7,4)(2,3,8,6)(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24);(9,13,15,11)(10,18,17,16)(12,22,23,19)(14,24,21,20)(25,28,30,37)(26,32,29,36)(27,35,34,40)(31,38,33,39);(1,6,7,3)(2,4,8,5)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,29,30,26)(27,31,34,33)(28,36,37,32)(35,38,40,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095196);