Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511571 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,22)(19,24,29,37)(20,25,32,38)(21,26,36,35)(27,39,44,43)(28,33,41,46)(30,40,47,45)(31,34,42,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,22,37)(20,30,32,47)(21,33,36,46)(25,40,38,45)(26,41,35,28)(27,42,44,31)(34,39,48,43);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20,23,32)(18,25,22,38)(19,27,29,44)(21,34,36,48)(24,39,37,43)(26,42,35,31)(28,45,41,40)(30,46,47,33);(3,8)(5,6)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418772);