Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138421 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19283 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18,21,27,22,28,36,43,23,25,37,44,30,39,20,26)(19,29,33,40,34,45,31,46,32,47,35,24,38,41,48,42);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19,30,34,23,32,22,38)(18,24,39,42,25,40,28,46)(20,35,37,48,36,33,21,31)(26,45,44,47,43,41,27,29);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(18,25)(20,36)(21,37)(24,40)(28,39)(31,48)(33,35)(42,46)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45655);