Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C2×#41 | M16 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,9,7,4,5,15)(3,10,13,11,14,16,12,8)(17,18,21,26,23,25,36,40,22,27,35,41,20,33,34,46)(19,28,31,43,32,38,45,47,29,42,44,48,30,24,37,39);(1,3,5,12,7,14,6,13)(2,8,9,10,4,11,15,16)(17,19,20,30,22,29,23,32)(18,24,25,28,27,38,33,42)(21,31,34,37,35,44,36,45)(26,39,40,43,41,47,46,48);(1,4)(2,7)(3,11)(5,15)(6,9)(8,14)(10,13)(12,16)(19,29)(24,38)(28,42)(30,32)(31,44)(37,45)(39,47)(43,48)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^c=egi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41914);