Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C8×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,12)(8,17)(9,19)(10,14)(11,15)(13,18)(16,20)(21,22,26,30,27,29,34,35,25,23,28,32,24,31,33,36);(1,2,3,4)(5,7,9,15,10,16,18,8)(6,11,13,12,14,17,19,20)(21,23,26,32,27,31,34,36,25,22,28,30,24,29,33,35);(5,8,18,16,10,15,9,7)(6,12,19,11,14,20,13,17)(21,24,25,27)(22,29,23,31)(26,33,28,34)(30,35,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47907);