Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132181 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #35687 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#548 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1387 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,3,6,4,5,7,8)(10,14,16,19)(15,23,24,22);(2,5)(9,10)(11,15)(12,19)(13,16)(14,21)(17,22)(18,24)(20,23);(2,5)(9,11,13,18)(10,15,16,24)(12,20,21,17)(14,22,19,23)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72268912 |
| maximal quotient of | #72268841 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39545);