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 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,14,16)(12,22,23,18)(13,19,24,21)(25,26)(27,32)(28,33)(29,35)(30,31)(34,39)(36,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,22,24,18)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,39)(37,40);(2,6)(3,7)(10,17)(13,24)(16,20)(18,22)(26,31)(28,38)(32,40)(34,35);(3,7)(4,8)(9,12)(10,18)(11,21)(13,16)(14,19)(15,23)(17,22)(20,24)(25,27,30,37)(26,32,31,40)(28,34,38,35)(29,33,39,36);(1,3)(2,4)(5,7)(6,8)(9,13)(10,19)(11,18)(12,20)(14,22)(15,24)(16,23)(17,21)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4)(2,3)(5,8)(6,7)(9,14,15,11)(10,20,17,16)(12,21,23,19)(13,22,24,18)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971329);