Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1361 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 156 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) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(6,12)(7,17)(8,18)(9,10)(13,25)(14,30)(15,31)(16,28)(19,35)(20,34)(21,32)(22,33)(23,24)(26,27)(29,36);(5,6,7,13,18,29,35,27,10,15,20,28,24,14,32,30,11,16,21,31,23,26,34,36,9,25,19,12,8,22,17,33)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,e^2=fgi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1556);