Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×C22⋊C16 |
| prime radical | #532 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,9,15,10,16,28,41,11,17,29,42,31,44,56,64,12,18,30,43,32,45,57,65,8,14,20,34,26,39,49,59)(7,19,22,46,23,47,51,66,21,37,48,40,50,61,67,63,24,38,27,54,52,62,55,68,25,13,33,35,53,36,58,60);(5,7,20,48,32,52,56,58,11,21,9,27,26,53,57,51,12,24,29,33,10,23,49,67,8,25,30,22,31,50,28,55)(6,13,34,46,45,61,64,68,17,19,15,40,39,62,65,60,18,37,42,54,16,36,59,66,14,38,43,35,44,47,41,63);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^b=di,f^b=fi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60075);