Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #62428 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2965 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,4,7,5,3,8,6)(9,10,12,15,22,19,23,24,13,16,21,20,14,11,17,18);(1,3)(2,5)(4,6)(7,8)(9,11)(10,14)(12,20)(13,19)(15,21)(16,22)(17,18)(23,24);(1,2)(3,5)(4,6)(7,8)(12,21)(15,20)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=eg,f^c=fgi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38872);