Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
620 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 52 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,26,32,28,31,40,48,27,33,39,49,25,41,24,30)(23,34,37,44,36,29,43,45,38,47,51,52,42,46,35,50);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,27,38)(22,29,33,46)(24,35,40,43)(25,42,28,36)(26,37,39,51)(30,44,48,52)(31,47,41,34)(32,45,49,50);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,25,39,27,40,28,26)(22,30,41,49,33,48,31,32)(23,35,42,51,38,43,36,37)(29,44,34,50,46,52,47,45)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,f^b=fgi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36582);