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 | |
4473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,12,19,16)(13,17,22,23)(18,20,21,24)(25,26,31,34)(27,32,30,28)(29,37,39,33)(35,36,38,40);(2,6)(3,7)(10,15)(12,16)(17,23)(20,24)(25,27,31,30)(26,32,34,28)(29,38,39,35)(33,36,37,40);(1,3)(2,4)(5,7)(6,8)(25,28)(26,30)(27,34)(29,36)(31,32)(33,38)(35,37)(39,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,19)(10,16,15,12)(13,21,22,18)(17,20,23,24)(25,28)(26,27)(29,36)(30,34)(31,32)(33,35)(37,38)(39,40);(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,29,31,39)(26,33,34,37)(27,35,30,38)(28,36,32,40);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30,31,27)(26,32,34,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349730);