Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
528 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,23,20,19,22,24);(1,2,4,7,5,8,3,6)(10,13)(11,16)(14,15)(18,19)(23,24);(17,19,21,24,20,18,22,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4056);