Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501332 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13331 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 569 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(9,10,11,16,19,15,24,22,13,18,20,23,12,17,21,14);(1,2,4,5)(3,6,7,8)(9,11,19,24,13,20,12,21)(10,14,17,23,18,22,15,16);(2,5)(6,8)(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,22)(23,24);(1,3,4,7)(2,6,5,8)(11,20)(14,22)(16,23)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=egi,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72224853 |
| maximal normal subgroup of | #72224853 |
| maximal quotient of | #72224850 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415168);