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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22,31,38,43,32,19,23,34,29,16,21,30,17)(15,24,26,35,25,40,42,37,28,41,44,36,27,39,33,20);(5,7,8,11)(6,9,10,12)(13,15,19,28)(14,20,23,37)(16,25,31,27)(17,24,32,41)(18,33,34,42)(21,35,38,36)(22,39,29,40)(26,43,44,30);(6,10)(9,12)(13,16,19,31)(14,21,23,38)(15,25,28,27)(17,29,32,22)(18,30,34,43)(20,35,37,36)(24,40,41,39)(26,42,44,33);(13,17,30,21,16,29,34,23,19,32,43,38,31,22,18,14)(15,20,33,39,27,36,44,41,28,37,42,40,25,35,26,24)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404967);