Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #252235 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2598 |
| prime radical | C22×D4 | #41 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
510 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(1,2,4,7)(3,8,5,6)(9,10)(11,18)(12,15)(13,17)(14,19)(16,21)(20,23)(22,24);(1,3,4,5)(2,6,7,8)(9,11,12,19,13,20,21,24)(10,14,16,18,17,22,15,23);(2,6)(3,5)(7,8)(14,22)(18,23);(1,3,4,5)(2,6,7,8)(10,15,17,16)(14,23,22,18)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=di,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72307197, #72307212, #72307229, … |
| maximal normal subgroup of | #72307197, #72307212, #72307229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 233827);