Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4414 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)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,5)(6,8)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,37,38,39);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(27,34)(30,38)(33,35)(37,39);(10,16)(14,17)(15,24)(22,23)(28,36)(31,40)(33,35)(37,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608520);