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 | #41372 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4409 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)(10,14)(11,17)(12,19)(22,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,15,17,21)(12,20,19,16)(18,24,23,22)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,8,6)(9,12,13,19)(10,16,14,20)(11,18,17,23)(15,22,21,24)(27,34)(30,38)(33,35)(37,39);(11,17)(15,21)(18,23)(22,24)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=g,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, 7547451);