Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133889 |
| 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×#117 | C23 |
| prime radical | #35514 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1018 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)(9,10,14,18,15,20,12,22)(11,19,21,16,13,17,23,24);(2,5)(3,7)(9,11)(10,16)(12,23)(13,15)(14,21)(17,18)(19,22)(20,24);(10,17,20,19)(12,14)(16,18,24,22)(21,23)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72278405, #72278562, #72278949 |
| maximal normal subgroup of | #72278405, #72278562, #72278949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41295);