Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93550 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1295 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | C4×#97 | C22 |
| prime radical | C23 | #343 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C4×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,11,15,19,24,21,16,17,23,18,12,20,14,22);(2,5)(3,8)(6,7)(9,11,14,19,15,22,13,10,16,18,24,20,12,21,23,17)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=fgh,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 846);