Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4313 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,18)(13,19)(14,17)(16,21)(20,23)(22,24)(25,26,29,31)(27,33,34,30)(28,36,37,32)(35,38,40,39);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,18,21,16)(13,17,14,19)(20,23,22,24)(25,27,29,34)(26,30,31,33)(28,35,37,40)(32,38,36,39);(9,12)(10,16)(11,18)(13,20)(14,22)(15,21)(17,23)(19,24)(26,31)(27,34)(32,36)(35,40);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,17,11,19)(12,20,21,22)(16,23,18,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,3,5,4)(2,7,8,6)(9,14,15,13)(10,17,11,19)(12,20,21,22)(16,24,18,23);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548561);