Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | C23 | #493 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,22)(12,28)(13,31)(14,32)(15,21)(16,24)(17,25)(18,30)(19,23)(20,29)(26,40)(27,33)(34,37)(35,39)(36,38);(1,2,3,7,4,6,8,5)(9,11,23,37,14,26,12,24,15,27,30,39,29,38,13,25)(10,16,31,40,20,34,18,22,21,17,28,36,32,35,19,33);(9,11,23,37,14,26,12,24,15,27,30,39,29,38,13,25)(10,17,31,36,20,35,18,33,21,16,28,40,32,34,19,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=ghi,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48398);