Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22,25,28)(23,29,32,36)(24,33,30,27)(26,34,35,31);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16);(5,7,9,14)(6,12,13,19)(8,15,11,18)(10,16,17,20)(21,23,30,34,25,32,24,31)(22,26,33,36,28,35,27,29);(1,3,4,2)(7,14)(12,19)(15,18)(16,20)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=fhi,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120289);