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 | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,3)(5,6,8,7)(9,10,13,17)(11,19,21,38)(12,23,26,37)(14,28,25,18)(15,30,31,20)(16,33,35,27)(22,39,29,34)(24,36,40,32);(1,3,4,2)(6,7);(9,11,13,21)(10,15,17,31)(12,24,26,40)(14,29,25,22)(16,34,35,39)(18,36,28,32)(19,37,38,23)(20,33,30,27);(5,7,8,6)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fh,h^c=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405499);