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 |
| prime radical | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(13,21)(18,22)(19,24)(20,23)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,31,30)(26,28,32,34)(29,35,39,38)(33,36,37,40);(1,3,5,4)(2,7,6,8)(10,15)(12,16)(18,22)(20,23)(26,32)(28,34)(33,37)(36,40);(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,31,34)(26,27,32,30)(29,36,39,40)(33,35,37,38);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,39)(26,33,32,37)(27,35,30,38)(28,36,34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30,31,27)(26,34,32,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=hi,e^2=ghi,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, 7619006);