Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
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 | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,17,12,18,11,25)(7,13,16,28,20,27,15,9)(8,19,23,35,24,31,29,34)(14,26,32,30,33,36,21,22);(1,2,3,4)(5,7)(6,13)(8,21)(9,25)(10,16)(11,15)(12,20)(14,23)(17,28)(18,27)(19,22)(24,32)(26,35)(29,33)(30,31)(34,36);(5,8)(6,14)(7,19)(9,26)(10,23)(11,29)(12,24)(13,30)(15,34)(16,35)(17,32)(18,33)(20,31)(21,25)(22,27)(28,36)Polycyclic presentation:
2^9|a^2=g,b^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, 55296);