Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
642 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)(5,6,9,11,10,14,8,13)(7,15,17,20,18,12,16,19)(21,22,25,28)(23,29,32,36)(24,33,30,27)(26,34,35,31);(1,3,4,2)(6,11)(7,16)(8,9)(13,14)(17,18)(19,20);(21,23,30,34,25,32,24,31)(22,26,33,36,28,35,27,29);(5,7,8,16,10,18,9,17)(6,12,13,20,14,15,11,19)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=de,e^b=eh,g^b=gh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138777);