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 | #51522 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(12,20)(16,23)(19,21)(22,24)(26,31)(27,34)(29,38)(36,39);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(26,31)(28,33)(32,35)(36,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,26,30,31)(27,35,34,32)(28,29,33,38)(36,40,39,37);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,27,30,34)(26,32,31,35)(28,36,33,39)(29,37,38,40);(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,28)(26,29)(27,36)(30,33)(31,38)(32,37)(34,39)(35,40);(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)(25,29)(26,33)(27,37)(28,31)(30,38)(32,39)(34,40)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620684);