Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145382 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10473 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,15)(4,9)(5,12)(6,25)(7,26)(8,14)(10,21)(11,19)(13,23)(16,30)(17,27)(18,31)(20,28)(22,29)(24,32);(1,3)(2,9)(4,19)(5,21)(6,22)(7,20)(8,18)(10,13)(11,28)(12,15)(14,29)(16,23)(17,24)(25,30)(26,27)(31,32);(1,4,5,16,23,17,6,3,8,22,24,20,7,21,19,18)(2,10,11,28,26,29,32,15,14,27,25,30,13,9,12,31)Polycyclic presentation:
2^9|c^2=dh,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,h^b=hi,d^c=di,e^c=egh,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52578);