Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,29,30,34,40,39)(27,33,36,35,28,32,38,37);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,28)(26,32,34,33)(29,37,39,35)(31,36,40,38);(1,3,5,4)(2,6,7,8)(25,28,30,27)(26,33,34,32)(29,35,39,37)(31,38,40,36);(1,4,5,3)(2,6,7,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(27,28)(32,33)(35,37)(36,38);(10,16)(14,17)(15,24)(22,23)(25,29,31,34,30,39,40,26)(27,35,36,32,28,37,38,33)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6313852);