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 | |
4385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,37)(28,31)(29,34)(30,35)(32,36)(33,39)(38,40);(1,2,5,4)(3,7,8,6)(9,11,15,20)(10,16,17,24)(12,21,23,14)(13,19,18,22)(25,27,32,39)(26,33,36,37)(28,38,29,30)(31,35,34,40);(1,3,5,8)(2,6,4,7)(10,17)(11,20)(12,23)(19,22)(25,28,32,29)(26,31,36,34)(27,38,39,30)(33,35,37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,21)(13,17)(14,20)(15,23)(16,22)(19,24)(25,29,32,28)(26,34,36,31)(27,30,39,38)(33,40,37,35);(9,13,15,18)(10,12,17,23)(11,22,20,19)(14,24,21,16)(25,30)(26,35)(27,28)(29,39)(31,37)(32,38)(33,34)(36,40);(9,14,15,21)(10,19,17,22)(11,12,20,23)(13,16,18,24)(25,31,32,34)(26,28,36,29)(27,35,39,40)(30,33,38,37)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=g,e^2=h,f^2=gi,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, 9624021);