Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150405 |
| 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 | #382 | C23 |
| prime radical | #3493 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 356 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(18,23,24,31,26,28,25,32)(20,27,30,29);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,31)(29,32);(2,7)(6,14)(8,15)(9,16)(17,19,22,21)(20,29,30,27)Polycyclic presentation:
2^9|a^2=e,c^2=hi,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cgi,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57976);