Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#97 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,27,22,30,21,28,24,29,36,44,23,26,37,43)(19,25,32,39,35,41,33,40,34,38,46,47,31,42,45,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,33,23,31,20,32,24,34,37,45,22,35,36,46)(18,25,28,40,26,42,27,39,29,38,43,48,30,41,44,47);(3,9)(6,12)(10,15)(13,16)(18,26,29,30)(19,31,34,35)(20,36)(21,37)(25,38)(27,28,44,43)(32,33,46,45)(41,42)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,h^2=i,c^a=chi,f^a=fi,c^b=cg,f^b=fi,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33737);