Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #384 | C23 |
| prime radical | #537 | C2×C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,20,22,29,23,33,26,15)(12,16)(13,17)(14,19)(18,27)(21,34,35,39,36,40,37,28)(24,30)(25,32)(31,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,26)(14,23)(16,28)(17,20)(18,33)(19,29)(22,27)(24,37)(25,36)(30,34)(31,40)(32,39)(35,38);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,19,32)(11,21,23,36)(13,24,27,38)(15,28,29,39)(17,30,18,31)(20,34,33,40)(22,35,26,37)Polycyclic presentation:
2^9|a^2=e,c^2=gi,d^2=g,e^2=hi,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57912);