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 | #39137 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4470 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)(4,6)(5,8)(9,10)(11,19)(12,14)(13,16)(15,17)(18,23)(20,24)(21,22)(25,26,30,33)(27,28,31,35)(29,32,38,40)(34,36,37,39);(1,2,4,6)(3,5,8,7)(9,11,15,13)(10,16,17,19)(12,20,23,22)(14,21,18,24)(27,31)(28,35)(34,37)(36,39);(1,2)(3,5)(4,6)(7,8)(10,17)(14,18)(16,19)(21,24)(25,27,30,31)(26,28,33,35)(29,37,38,34)(32,39,40,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,20)(13,22)(14,17)(15,23)(16,24)(19,21)(25,28)(26,31)(27,33)(29,36)(30,35)(32,37)(34,40)(38,39);(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)(27,34,31,37)(28,36,35,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^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, 8103606);