Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #385 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
377 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 92 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)(10,13)(11,15)(14,16)(17,18,21,19,20,22,24,23);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,19,21,22,20,23,24,18);(1,2)(3,6)(4,8)(5,7)(11,15)(13,16)(18,22)(19,23)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=ci,d^b=dg,e^b=eh,d^c=dh,e^c=egh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6527);