Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,14,10,16)(7,17,20,31,19,12,25,28)(8,21,24,32,23,13,26,30)(18,27,33,35,22,29,34,36);(5,7,10,20,11,19,9,25)(6,12,16,28,14,17,15,31)(8,22,24,33,23,18,26,34)(13,29,30,35,21,27,32,36);(2,4)(5,8)(6,13,14,21)(7,18)(9,24)(10,26)(11,23)(12,27,17,29)(15,30,16,32)(19,22)(20,33)(25,34)(28,35,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30975);