Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2654 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 2654 conjugacy classes of subgroups — too many to draw. Only its 306 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 — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,15,16,13)(12,14)(17,18,20,24,21,22,19,23);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,20);(2,4)(17,19,21,20)(18,23,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=ci,f^b=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12711);