Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | #19125 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,9,12)(3,10,11,13)(17,18,19,23,27,31,32,30,21,25,28,29,20,24,26,22);(1,2,7,6,5,9,4,12)(3,8,10,14,11,16,13,15)(18,22)(19,26)(20,27)(23,24)(25,30)(28,32)(29,31);(1,3)(2,8)(4,13)(5,11)(6,14)(7,10)(9,16)(12,15)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44241);