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)(11,15)(12,21)(13,23)(14,16)(17,22)(18,19)(20,24)(26,31)(29,39)(32,38)(33,35);(1,2,5,6)(3,7,4,8)(9,11,14,20)(10,15,16,24)(12,19,22,13)(17,23,21,18)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,38)(39,40);(10,16)(15,24)(17,21)(18,23)(26,31)(28,37)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,20)(14,22)(15,23)(16,21)(18,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,39,33);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,28,30,37)(26,33,31,35)(27,34,36,40)(29,38,39,32);(9,13,14,19)(10,18,16,23)(11,12,20,22)(15,17,24,21)(25,29)(26,34)(27,35)(28,38)(30,39)(31,40)(32,37)(33,36)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549825);