Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| 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 | |
488 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)(5,6,10,17,12,18,11,29)(7,13,15,9,20,27,16,28)(8,21,24,34,25,19,30,33)(14,26,22,35,32,36,31,23);(1,2,4,3)(5,7)(6,13)(8,22)(9,17)(10,15)(11,16)(12,20)(14,30)(18,27)(19,23)(21,35)(24,32)(25,31)(26,33)(28,29)(34,36);(1,3,4,2)(5,8)(6,14)(7,19)(9,26)(10,24)(11,30)(12,25)(13,23)(15,33)(16,34)(17,22)(18,32)(20,21)(27,35)(28,36)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55956);