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 | #42518 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4426 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)(3,6)(4,8)(5,7)(9,10,15,18)(11,16,13,23)(12,14,22,17)(19,24,21,20)(26,31)(27,34)(28,36)(37,39);(2,3)(7,8)(9,11,15,13)(10,16,18,23)(12,21,22,19)(14,20,17,24)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(2,3)(7,8)(9,12,15,22)(10,17,18,14)(11,19,13,21)(16,20,23,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,3,6,2)(4,8,5,7)(9,12)(10,14)(11,19)(13,21)(15,22)(16,24)(17,18)(20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4,6,5)(2,7,3,8)(9,13,15,11)(10,16,18,23)(12,21,22,19)(14,24,17,20)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,5,6,4)(2,8,3,7)(9,14)(10,12)(11,20)(13,24)(15,17)(16,21)(18,22)(19,23)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014865);