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 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,40)(35,38);(1,2,5,4)(3,7,8,6)(10,15)(12,17)(16,22)(20,23)(25,27,30,33)(26,28,31,36)(29,38,39,34)(32,40,37,35);(2,4)(6,7)(13,21)(16,22)(18,24)(20,23)(26,31)(27,33)(32,37)(34,38);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,30,39)(26,32,31,37)(27,34,33,38)(28,35,36,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161922);