Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111399 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1088 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,4,6)(5,8)(9,10,14,19,15,20,13,18)(11,21,24,16,25,31,23,32)(12,17,27,34,28,36,29,35)(22,33,26,30,37,39,38,40);(1,3)(2,6)(4,5)(7,8)(9,11,15,25)(10,16,20,32)(12,26,28,38)(13,24,14,23)(17,33,36,39)(18,31,19,21)(22,29,37,27)(30,34,40,35);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,35)(19,34)(20,36)(21,33)(23,38)(24,26)(25,37)(31,39)(32,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18600);