Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
611 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(3,7)(9,10,14,17)(11,18)(12,16,13,22)(15,19)(20,24)(21,23);(2,5)(3,7)(6,8)(11,19,20,21)(15,18,23,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,24)(17,23)(18,22)Polycyclic presentation:
2^9|a^2=dg,b^2=d,e^2=g,f^2=gi,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,e^b=eh,f^b=fi,d^c=dg,e^c=eg,f^c=fgi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233386, #72233387, #72233402, … |
| maximal normal subgroup of | #72233386, #72233387, #72233402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51816);