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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4122 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)(3,5)(4,6)(7,8)(9,10,15,13)(11,19,22,16)(12,23,21,17)(14,18,20,24)(25,26)(27,33)(28,38)(29,31)(30,34)(32,35)(36,39)(37,40);(1,3,7,6)(2,4,8,5)(9,11)(10,16)(12,20)(13,19)(14,21)(15,22)(17,24)(18,23)(25,27,32,29)(26,33,35,31)(28,37,30,36)(34,39,38,40);(1,4,7,5)(2,3,8,6)(25,28,32,30)(26,34,35,38)(27,36,29,37)(31,39,33,40);(2,8)(4,5)(9,12,15,21)(10,17,13,23)(11,20,22,14)(16,24,19,18)(25,29,32,27)(26,31,35,33)(28,36,30,37)(34,40,38,39);(1,5,7,4)(2,3,8,6)(9,13,15,10)(11,19,22,16)(12,17,21,23)(14,18,20,24)(25,30,32,28)(26,34,35,38)(27,37,29,36)(31,39,33,40);(1,6,7,3)(2,5,8,4)(9,14,15,20)(10,18,13,24)(11,21,22,12)(16,23,19,17)(25,31)(26,27)(28,39)(29,35)(30,40)(32,33)(34,36)(37,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10054765);