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 | |
4508 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2,5,6)(3,7,8,4)(9,11,15,20)(10,16,17,23)(12,19,22,14)(13,18,24,21)(26,32)(27,28)(34,37)(35,36);(2,6)(4,7)(10,17)(11,20)(14,19)(18,21)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(2,6)(3,8)(9,12)(10,18)(11,19)(13,23)(14,20)(15,22)(16,24)(17,21)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,19)(11,21)(12,16)(14,17)(15,24)(18,20)(22,23)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,19)(10,13,17,24)(11,12,20,22)(16,18,23,21)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295994);