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 | |
502 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)(5,6,11,17,9,15,12,18)(7,16,10,26,20,13,21,29)(8,22,23,32,25,36,24,28)(14,27,30,19,31,34,33,35);(5,7,12,21,9,20,11,10)(6,13,18,26,15,16,17,29)(8,19,23,34,25,35,24,27)(14,28,30,22,31,32,33,36);(5,8,11,24,9,25,12,23)(6,14,18,30,15,31,17,33)(7,19,10,27,20,35,21,34)(13,28,26,22,16,32,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,e^b=ei,d^c=di,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, 54098);