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 | |
4510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,19)(12,14)(13,16)(15,17)(18,23)(20,24)(21,22)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,39,37,36);(2,5)(6,7)(9,11,15,13)(10,16,17,19)(12,20,23,22)(14,21,18,24)(27,34)(28,31)(35,37)(36,39);(1,2)(3,6)(4,5)(7,8)(10,17)(14,18)(16,19)(21,24)(25,27)(26,31)(28,33)(29,37)(30,34)(32,36)(35,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,18)(11,20)(13,22)(14,17)(15,23)(16,24)(19,21)(25,28,30,31)(26,27,33,34)(29,36,38,39)(32,35,40,37);(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,24,18,21);(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,20,19,22)(25,29,30,38)(26,32,33,40)(27,35,34,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=ghi,e^2=h,f^2=gi,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, 8104091);