Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C8×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,18,22,19,15,23,25,17,21,26,28,16,24,20,27);(2,4)(6,10)(9,12)(13,15,18,25,19,21,23,28,17,24,26,27,16,14,20,22);(5,6,7,9,8,10,11,12)(13,16,17,19)(14,15,21,24)(18,20,26,23)(22,25,28,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=cgi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47547);