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 | #37498 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4348 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)(11,20)(12,22)(13,15)(14,24)(16,19)(17,23)(18,21)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,24)(14,17)(15,21)(16,22)(18,23)(19,20)(26,31)(27,34)(33,36)(35,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,21)(14,23)(15,24)(16,20)(17,18)(19,22)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,23)(10,17,19,15)(11,18,22,24)(12,21,20,14)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,14,16,21)(10,18,19,24)(11,17,22,15)(12,23,20,13);(1,2)(3,5)(4,7)(6,8)(9,15,16,17)(10,13,19,23)(11,21,22,14)(12,24,20,18)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329240);