Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×C4⋊C8 | C22 |
| socle | C22 | #1280 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
366 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,6,8,5)(4,7)(9,10,11,15,12,16,18,22,13,14,17,20,19,21,23,24);(2,5,7,6)(10,14)(16,21)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,d^b=dfhi,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 171);