Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 3 generators on 44 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,14,18,23,20,25,17,33)(15,26,28,21,30,36,34,38)(16,22,19,24,32,39,31,40)(27,41,29,35,42,37,43,44);(5,7)(6,9,10,11)(8,12)(13,15,19,29,18,28,32,42,20,30,31,43,17,34,16,27)(14,21,24,37,23,36,39,44,25,38,40,41,33,26,22,35);(1,3,4,2)(13,16,17,31,20,32,18,19)(14,22,33,40,25,39,23,24)(15,27,34,43,30,42,28,29)(21,35,26,41,38,44,36,37)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fgi,d^b=dg152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35529);