Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10048 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | 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 | |
434 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(3,11)(4,16)(5,19)(7,22)(8,17)(10,21)(12,30)(13,18)(14,29)(15,28)(20,31)(23,25)(24,27)(26,32);(1,2)(3,12,14,7)(4,17)(5,21)(6,9)(8,16)(10,19)(11,27,29,25)(13,23,15,24)(18,30,28,22)(20,32)(26,31);(1,3)(2,7)(4,18)(5,13)(6,14)(8,25)(9,12)(10,23)(11,16)(15,19)(17,22)(20,28)(21,24)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50456);