Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | |
768 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)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(21,22,25,28)(23,29,32,36)(24,33,30,27)(26,34,35,31);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20);(7,14)(10,19)(13,20)(15,18)(21,23,30,34,25,32,24,31)(22,26,33,36,28,35,27,29);(1,3,4,2)(7,14)(10,19)(13,20)(15,18)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120313);