Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #102 |
| commutator | C8 | C82 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C22 | #1279 |
| prime radical | C23 | #339 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,14)(12,16)(13,18)(15,21)(17,20)(19,23)(22,24);(10,14,16,15,18,23,17,22)(11,19)(21,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,h^a=hi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216227, #72216228, #72216234, … |
| maximal normal subgroup of | #72216227, #72216228, #72216234, … |
| maximal quotient of | #72216538, C2×#93266 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 462);