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 | |
4235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(26,31)(27,33)(34,39)(35,40);(1,2,5,8)(3,4,7,6)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(10,16)(12,17)(15,22)(18,23);(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,4)(2,7)(3,8)(5,6)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|b^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9582118);