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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,17)(13,16)(15,19)(18,22)(20,24)(21,23)(25,26)(27,33)(28,30)(29,34)(31,32)(35,40)(36,37)(38,39);(1,2,4,5)(3,6,7,8)(9,10,14,11)(12,15,19,17)(13,18,22,16)(20,24,21,23)(26,32)(27,30)(34,39)(35,37);(10,11)(12,19)(16,18)(20,21)(25,27,31,30)(26,33,32,28)(29,37,38,35)(34,36,39,40);(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,28)(26,27)(29,36)(30,32)(31,33)(34,35)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29,31,38)(26,34,32,39)(27,35,30,37)(28,36,33,40);(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,30,31,27)(26,28,32,33)(29,35,38,37)(34,40,39,36)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893544);