Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149897 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #3484 | C4 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,8,12,16,29)(7,11,19,27,17,26,34,35)(10,15,24,32,20,13,22,25)(18,28,21,31,23,30,33,36);(5,7,16,33,8,17,9,21)(6,11,25,27)(10,23,24,19,20,18,22,34)(12,26,32,35)(13,30,14,31)(15,28,29,36);(2,4)(6,12)(7,17)(13,15)(14,29)(18,23)(19,34)(21,33)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dgi,f^b=fgi,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59860);