Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94427 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C22×Q8 | C42 |
| Frattini, cosocle | C4×#124 | C22 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(1,2,3,6,4,5,7,8)(9,10,12,17,13,19,14,18)(11,15,20,23,21,16,22,24);(2,5)(3,7)(10,15,18,24,19,16,17,23)(11,20,21,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=eg,g^a=gi,h^a=hi,c^b=cg,d^b=dfi,g^b=gh,e^c=ehi,g^c=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1663);