Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6,7,8)(10,16)(12,21)(13,27)(14,29)(17,24)(18,36)(19,33)(20,30)(22,38)(25,26)(28,31)(32,37)(34,40)(35,39);(9,10,15,19)(11,17,23,35)(12,24,26,39)(13,28,30,36)(14,31,29,18)(16,20,33,27)(21,32,25,40)(22,37,38,34);(1,2,3,4)(9,11,15,23)(10,17,19,35)(12,20,26,27)(13,21,30,25)(14,22,29,38)(16,24,33,39)(18,34,31,37)(28,32,36,40);(9,12,15,26)(10,17,19,35)(11,20,23,27)(13,22,30,38)(14,21,29,25)(16,32,33,40)(18,37,31,34)(24,36,39,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405081);