Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #237997 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#393 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1087 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 190 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 — 4 generators on 24 points:
(2,5)(3,7)(6,8)(11,18,20,21)(15,19,23,24);(9,10)(11,19,20,24)(12,22)(13,16)(14,17)(15,21,23,18);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,21)(13,18)(14,20)(16,24)(17,23)(19,22);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,c^b=cg,f^b=fh,d^c=di,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280510, #72280527, #72280537, … |
| maximal normal subgroup of | #72280510, #72280527, #72280537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145193);