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 | #59215 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26)(27,31)(28,33)(29,37)(30,32)(34,38)(35,36)(39,40);(1,2,5,4)(3,7,8,6)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,39,37,36);(2,4)(6,7)(10,17)(14,21)(16,18)(20,24)(26,32)(27,33)(34,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,40,37,35);(9,13,15,11)(10,18,17,16)(12,22,23,19)(14,24,21,20)(25,29,30,38)(26,34,32,37)(27,35,33,40)(28,36,31,39);(1,4,5,2)(3,7,8,6)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=gh,e^2=h,f^2=ghi,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, 8103110);