Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1732 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1732 conjugacy classes of subgroups — too many to draw. Only its 182 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)(11,18,20,23)(12,16)(13,22)(14,17)(15,21,24,19);(1,3)(2,5)(4,7)(6,8)(11,19,20,21)(15,18,24,23);(5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,24)(18,22);(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255823, #72256011, #72256044, … |
| maximal normal subgroup of | #72255823, #72256011, #72256044, … |
| maximal quotient of | #72261684, #72263332, #72266591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145229);