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) = C24 |
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 | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,29)(18,33)(19,23)(20,26)(22,31)(25,41)(27,42)(28,39)(30,35)(32,40)(34,38)(36,44)(37,43);(1,3,4,2)(6,9)(7,11);(13,15,19,27)(14,20,23,37)(16,25,18,34)(17,30,32,44)(21,35,33,36)(22,38,39,41)(24,40,42,29)(26,28,43,31);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,33)(15,25,27,34)(17,28,32,31)(20,35,37,36)(22,29,39,40)(24,41,42,38)(26,30,43,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=ei,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, 405452);