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 | |
4652 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)(3,8)(4,7)(5,6)(9,10)(11,19)(12,17)(13,23)(14,15)(16,20)(18,24)(21,22)(26,31)(27,32)(33,38)(34,39);(3,4)(7,8)(10,15)(12,22)(16,19)(18,23)(25,26,30,31)(27,28,32,35)(29,33,37,38)(34,36,39,40);(1,3)(2,7)(4,6)(5,8)(10,15)(11,20)(12,22)(13,24)(27,32)(28,35)(29,37)(33,38);(3,4)(7,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,19)(17,24)(18,22)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,4,6,3)(2,7,5,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,19,24,16)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(1,5,6,2)(3,7,4,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,36,35,40)Polycyclic presentation:
2^9|b^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886386);