Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1714 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1714 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,20)(16,19)(17,24)(18,22)(21,23);(1,2,3,4)(10,14)(11,17)(13,19)(15,23)(18,21)(22,24);(9,11)(10,15)(12,18)(13,21)(14,22)(16,23)(17,19)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=di,f^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,f^c=fi,h^c=hi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72278776, #72284236, #72284380, … |
| maximal normal subgroup of | #72278776, #72284236, #72284380, … |
| maximal quotient of | #72284504, #72284651, #72286370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55366);