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 | #2668 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(10,15)(12,19)(14,20)(18,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(10,15)(11,17)(16,22)(18,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,35)(30,33)(34,40)(38,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=h,f^2=g,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, 7547985);