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 | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(27,28)(31,32)(35,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(2,5)(6,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,3,4,7)(2,6,5,8)(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8040064);