Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1926 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,18,11,17)(7,14,22,33,23,30,27,34)(8,24,20,32,25,31,21,13)(9,26,19,15,29,35,28,36);(2,4)(5,7,11,22,12,23,10,27)(6,13,16,31,18,32,17,24)(8,19,20,9,25,28,21,29)(14,15,34,35,30,36,33,26);(5,8,12,25)(6,14)(7,19,23,28)(9,27,29,22)(10,21,11,20)(13,15)(16,34)(17,33)(18,30)(24,26)(31,35)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54488);