Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2087 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
478 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)(5,6,10,16,29,27,30,25,12,17,9,15,11,8,14,26)(7,18,22,34,32,36,28,13,21,19,20,24,23,35,31,33);(1,2,4,3)(5,7,11,32,12,21,29,23)(6,13,8,24,17,33,27,34)(9,28,30,20,10,31,14,22)(15,35,25,18,16,36,26,19);(1,3,4,2)(5,8,9,25,29,6,14,15,12,27,10,26,11,17,30,16)(7,19,20,34,32,35,31,13,21,18,22,24,23,36,28,33)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,h^b=hi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56266);