Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251935 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1762 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1762 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,21)(13,16)(14,20)(15,23)(18,22)(19,24);(1,3,4,7)(2,5,6,8)(9,10)(11,18,19,14)(12,15)(13,16)(17,20,24,22)(21,23);(5,8)(9,11)(10,14)(12,22)(13,19)(15,24)(16,18)(17,21)(20,23);(12,23)(15,21)(17,24)(20,22)Polycyclic presentation:
2^9|b^2=ei,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233376, #72233417, #72233462, … |
| maximal normal subgroup of | #72233376, #72233417, #72233462, … |
| maximal quotient of | #72233366, #72233386, #72233403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159147);