Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150578 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,4,10)(3,13,16,31)(5,21,15,28)(6,22,20,14)(7,8,17,24)(9,23,12,26)(11,18,29,30)(19,25,32,27);(1,2,5,11,7,8,16,25)(3,14,17,28,15,9,4,13)(6,22,19,31,23,12,30,21)(10,26,27,32,24,20,29,18);(2,8)(3,15)(4,17)(11,25)(12,22)(18,32)(20,26)(21,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gh,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efghi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ehi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72209805, #72227655, #72338942, … |
| maximal normal subgroup of | #72209805, #72227655, #72338942, … |
| maximal quotient of | #72227644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57825);