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 | C2×C8 | D16 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1331 |
| prime radical | C23 | #359 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2)(3,4)(5,6,9,17,12,20,31,19,14,21,11,40,13,41,37,52)(7,22,24,54,27,56,57,55,29,44,26,65,28,66,62,67)(8,30,32,18,34,49,39,51,35,50,33,16,10,38,36,48)(15,42,43,59,46,64,45,60,47,58,53,25,63,61,68,23);(2,4)(5,7,10,25,37,62,33,58,13,28,35,60,11,26,39,64,14,29,34,59,31,57,32,42,12,27,8,23,9,24,36,61)(6,15,18,44,40,53,38,54,20,46,51,66,52,68,30,55,21,47,16,22,17,43,49,65,41,63,48,56,19,45,50,67)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,f^2=hi,g^2=i,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, 1248);