Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | D16 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1380 |
| prime radical | C23 | #358 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,4,7,5,8,3,6)(9,10,13,20,16,23,17,41)(11,24,26,54,29,55,30,65)(12,21,32,49,33,51,34,66)(14,19,35,47,37,48,38,50)(15,39,36,52,40,67,31,22)(18,42,43,57,46,58,68,25)(27,53,59,70,61,71,62,45)(28,63,60,69,64,72,56,44);(2,6,8,7)(9,11,14,27,15,28,32,42,13,26,35,59,36,60,33,57,16,29,37,61,40,64,34,58,17,30,38,62,31,56,12,25)(10,18,21,44,22,45,50,65,41,68,66,72,67,71,48,55,23,46,51,69,52,70,47,54,20,43,49,63,39,53,19,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fgi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1241);