Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | #390 | C23 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18,20,14)(12,16,22,15)(13,17)(19,24,21,23);(9,11)(10,14)(12,21)(13,19)(15,18)(16,24)(17,23)(20,22);(11,19)(14,23)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=e,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,e^b=efi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338268, #72338280, #72338281, … |
| maximal normal subgroup of | #72338268, #72338280, #72338281, … |
| maximal quotient of | #72338325, #72338328, #72339715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56735);