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 |
| prime radical | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,8)(3,5,6,4)(10,14)(11,17)(16,19)(18,23)(26,31)(27,28)(34,37)(35,36);(1,3,7,6)(2,4,8,5)(10,14)(15,21)(16,19)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,4,7,5)(2,3,8,6)(9,10,13,14)(11,15,17,21)(12,19,20,16)(18,24,23,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(1,5,7,4)(2,3,8,6)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28,30,27)(26,33,31,32)(29,36,38,35)(34,40,37,39);(1,6,7,3)(2,4,8,5)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,38)(26,34,31,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=g,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, 8103217);