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 | C23 | C24×C4 |
| prime radical | #34909 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,21)(13,17)(14,16)(18,23)(19,24)(20,22)(25,26,30,34,29,31,28,33)(27,35,37,40,36,32,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24);(10,16)(11,18)(17,21)(19,20)(26,31)(27,36)(33,34)(37,38);(3,8)(5,7)(9,12,14,13)(10,17,16,21)(11,19,18,20)(15,22,23,24)(25,27,29,36)(26,32,31,35)(28,38,30,37)(33,40,34,39);(2,6)(5,7)(9,13,14,12)(10,17,16,21)(11,20,18,19)(15,22,23,24)(25,28,29,30)(26,33,31,34)(27,37,36,38)(32,39,35,40)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6317703);