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 | |
4549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(11,16)(17,21)(18,24)(25,26)(27,32)(28,37)(29,31)(30,34)(33,38)(35,36)(39,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,12,16,19)(13,21,23,17)(18,22,24,20)(25,27,31,35)(26,32,29,36)(28,30,38,40)(33,39,37,34);(1,3,5,4)(2,7,6,8)(10,15)(12,19)(17,21)(20,22)(26,29)(27,35)(28,38)(34,39);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,12,15,19)(13,18,23,24)(17,20,21,22)(25,28)(26,33)(27,30)(29,37)(31,38)(32,39)(34,36)(35,40);(9,12)(10,16)(11,15)(13,22)(14,19)(17,18)(20,23)(21,24)(25,29,31,26)(27,36,35,32)(28,37,38,33)(30,34,40,39);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,17,15,21)(11,18,16,24)(12,20,19,22)(25,30)(26,34)(27,28)(29,39)(31,40)(32,37)(33,36)(35,38)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7879703);