Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,18,12,15,30,19)(7,20,24,36,26,14,25,35)(8,27,22,13,28,34,23,33)(9,29,21,16,10,31,32,17);(2,4)(5,7,11,24,12,26,30,25)(6,13,18,34,15,33,19,27)(8,21,22,10,28,32,23,9)(14,17,35,29,20,16,36,31);(5,8)(6,14)(7,21)(9,25)(10,24)(11,22)(12,28)(13,17)(15,20)(16,33)(18,35)(19,36)(23,30)(26,32)(27,31)(29,34);(6,15)(13,33)(14,20)(16,17)(18,19)(27,34)(29,31)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,h^a=hi,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401490);