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 | |
4457 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)(9,10,15,17)(11,18,13,16)(12,14,21,24)(19,23,22,20)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,17,18)(12,19,21,22)(14,20,24,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(10,17)(11,13)(12,21)(20,23)(26,28)(32,34)(33,36)(39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,14)(11,19)(13,22)(15,21)(16,20)(17,24)(18,23)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,13,15,11)(10,18,17,16)(12,22,21,19)(14,23,24,20)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,4,5,3)(2,8,6,7)(9,14)(10,12)(11,20)(13,23)(15,24)(16,19)(17,21)(18,22)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617113);