Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1919 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,10,15,11,17,30,29,12,16,9,25,27,8,14,24)(7,18,21,23,22,33,28,32,20,35,19,34,31,36,26,13);(1,2,4,3)(5,7,11,31,12,20,27,22)(6,13,8,23,16,32,17,34)(9,26,14,19,10,28,30,21)(15,33,24,35,25,36,29,18);(1,3,4,2)(5,8,9,24,11,6,14,15,12,17,10,29,27,16,30,25)(7,18,19,23,22,33,26,32,20,35,21,34,31,36,28,13)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,h^b=hi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56293);