Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #50286 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(26,31)(32,34)(33,38)(39,40);(1,3,5,4)(2,6,8,7)(25,26,30,31)(27,34,37,32)(28,38,29,33)(35,39,36,40);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,30,37)(26,32,31,34)(28,36,29,35)(33,40,38,39);(1,4,5,3)(2,6,8,7)(11,18)(14,17)(19,24)(21,23)(25,28,30,29)(26,33,31,38)(27,35,37,36)(32,39,34,40);(10,15)(14,17)(16,22)(21,23)(25,29,30,28)(26,33,31,38)(27,36,37,35)(32,39,34,40);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615050);