Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| 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 | |
294 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 114 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)(5,6,9,15,10,16,27,38,11,14,28,33,30,37,53,56,12,17,29,39,31,40,54,60,8,25,19,44,26,52,45,64)(7,18,21,41,22,42,47,61,23,36,48,58,50,59,65,67,24,43,49,62,51,63,66,68,20,13,32,34,46,35,55,57);(5,7,19,32,31,51,53,65,11,23,9,21,26,46,54,66,12,24,28,48,10,22,45,55,8,20,29,49,30,50,27,47)(6,13,33,41,40,59,64,68,14,18,39,58,52,63,38,57,17,36,44,62,16,35,56,61,25,43,15,34,37,42,60,67);(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^a=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60073);