Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501052 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33190 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,13,23,10,16,26,29,12,8,9,15,11,17,25,24)(7,14,28,34,27,31,35,33,21,22,19,30,20,18,32,36);(1,2,4,3)(5,7)(6,14)(8,22)(9,19)(10,27)(11,20)(12,21)(13,28)(15,30)(16,31)(17,18)(23,34)(24,36)(25,32)(26,35)(29,33);(5,8,12,6)(7,18,21,31)(9,24,13,29)(10,16,11,17)(14,27,22,20)(15,26,23,25)(19,34,28,30)(32,36,35,33);(1,3,4,2)Polycyclic presentation:
2^9|a^2=fi,b^2=e,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408845);