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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,20)(12,17,22,24)(13,18,23,21)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,20)(18,22)(23,24)(26,32)(27,28)(34,37)(35,36);(2,3)(6,7)(10,16)(11,19)(17,24)(18,21)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(4,8)(6,7)(9,12,14,22)(10,17,16,24)(11,18,19,21)(13,15,23,20)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3,5,2)(4,7,8,6)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,23)(22,24)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,17)(12,20)(14,23)(15,22)(16,21)(19,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520310);