Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #35293 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1104 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(1,2,4,7,5,3,8,6)(9,10)(11,18,20,14)(12,15)(13,16)(17,22)(19,24,21,23);(1,3)(2,5)(4,6)(7,8)(9,11)(10,14)(12,21)(13,20)(15,24)(16,18)(17,23)(19,22);(1,2)(3,5)(4,7)(6,8)(10,15)(12,22)(14,23)(16,17)(18,24)(19,21)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224698, #72224701, #72224702, … |
| maximal normal subgroup of | #72224698, #72224701, #72224702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18425);