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 | #39165 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4422 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)(10,15)(11,19)(13,23)(20,22)(25,26)(27,29)(28,32)(30,31)(33,38)(34,36)(35,40)(37,39);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,17,19,16)(12,13,18,23)(20,21,22,24)(26,31)(27,33)(32,39)(34,35);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,16,15,17)(12,22,18,20)(13,21,23,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,12,14,18)(10,13,15,23)(11,20,19,22)(16,24,17,21);(1,4,5,3)(2,8,6,7)(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,22)(23,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,13)(10,18)(11,21)(12,15)(14,23)(16,20)(17,22)(19,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=h,e^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, 7616694);