Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,15,18)(11,19,23,16)(12,13,17,14)(20,22,24,21)(25,26,30,34)(27,31,32,28)(29,33,39,40)(35,38,37,36);(1,2,5,3)(4,7,8,6)(9,11,15,23)(10,16,18,19)(12,20,17,24)(13,21,14,22)(25,27)(26,31)(28,34)(29,37)(30,32)(33,36)(35,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,20)(14,18)(15,17)(16,21)(19,22)(23,24)(25,28,30,31)(26,32,34,27)(29,38,39,36)(33,35,40,37);(2,3)(6,7)(9,13,15,14)(10,12,18,17)(11,21,23,22)(16,20,19,24)(25,29,30,39)(26,33,34,40)(27,35,32,37)(28,36,31,38);(1,4,5,8)(2,6,3,7)(9,13,15,14)(10,17,18,12)(11,22,23,21)(16,20,19,24)(26,34)(27,32)(33,40)(35,37);(9,14,15,13)(10,12,18,17)(11,22,23,21)(16,20,19,24)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012828);