Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150407 |
| 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 | C23 | C2×#99 |
| 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 | |
334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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)(11,20,23,31,24,35,29,15)(12,25)(13,17)(14,19)(16,27)(18,21)(22,32,26,39,37,38,40,30)(28,33)(34,36);(1,2,5,8)(3,6,4,7)(9,11,21,23,14,24,13,29)(10,15,17,20,19,31,18,35)(12,26,36,37,27,40,28,22)(16,32,34,39,25,38,33,30);(1,3,5,4)(2,7,8,6)(9,12)(10,16)(11,22,24,37)(13,28)(14,27)(15,30,31,39)(17,34)(18,33)(19,25)(20,32,35,38)(21,36)(23,26,29,40)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=gi,d^2=gh,e^2=hi,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efghi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,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, 58129);