Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
22618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(5,6,7,8)(10,16)(11,17)(14,28)(18,25)(20,30)(23,36)(26,38)(35,40);(1,2)(3,4)(5,6)(7,8)(9,10,15,16)(11,12,17,24)(13,25,27,18)(14,19,28,29)(20,33,30,22)(21,23,31,36)(26,37,38,32)(34,40,39,35);(9,11)(10,12)(13,20)(14,21)(15,17)(16,24)(18,22)(19,23)(25,33)(26,34)(27,30)(28,31)(29,36)(32,35)(37,40)(38,39);(9,12)(10,17)(11,16)(13,22)(14,23)(15,24)(18,30)(19,31)(20,25)(21,29)(26,35)(27,33)(28,36)(32,39)(34,37)(38,40);(9,13,15,27)(10,18,16,25)(11,20,17,30)(12,22,24,33)(14,26,28,38)(19,32,29,37)(21,34,31,39)(23,35,36,40);(9,14)(10,19)(11,21)(12,23)(13,26)(15,28)(16,29)(17,31)(18,32)(20,34)(22,35)(24,36)(25,37)(27,38)(30,39)(33,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532893);