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 | #1926 |
| 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 | |
450 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)(5,6,11,10,9,14,15,26)(7,12,18,27,20,17,19,30)(8,21,24,32,23,13,25,28)(16,29,33,36,31,34,35,22);(1,2,4,3)(5,7,15,19,9,20,11,18)(6,12,26,30,14,17,10,27)(8,22,24,29,23,36,25,34)(13,31,28,35,21,16,32,33);(1,3,4,2)(5,8)(6,13)(7,16)(9,23)(10,28)(11,24)(12,29)(14,21)(15,25)(17,34)(18,33)(19,35)(20,31)(22,30)(26,32)(27,36)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56155);