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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(26,28)(32,33)(34,36)(39,40);(3,8)(6,7)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(11,16)(13,21)(19,24)(25,27,31,29)(26,32,28,33)(30,37,38,35)(34,40,36,39);(3,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8816966);