Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139428 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9428 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #19727 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,25,4,11,8,14,21,10,16,12,5,22)(3,15,17,28,31,29,24,32,20,30,23,9,19,26,18,27);(1,3,16,31,8,20,7,19)(2,9,12,27,14,28,25,32)(4,17,21,18,5,23,6,24)(10,15,22,26,13,30,11,29);(1,4,8,5)(2,10,14,13)(3,17,20,23)(6,16,21,7)(9,15,28,30)(11,12,22,25)(18,19,24,31)(26,32,29,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=i,d^2=gh,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eg,f^c=fgh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72339870, #72339875, #72339876, … |
| maximal normal subgroup of | #72339870, #72339875, #72339876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46627);