Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | #1459 |
| prime radical | C2×#127 | C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
422 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 24 points:
(1,2)(3,6,8,5)(4,7)(9,10,12,15,13,16,21,22,14,17,11,18,19,23,20,24);(2,5)(3,8)(6,7);(9,11,13,20,14,12,19,21)(10,15,16,22,17,18,23,24)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=h,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dh,e^b=efh,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29578);