Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 48 points:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18,21,26,23,25,36,38,22,27,35,41,20,34,29,44)(19,28,32,43,33,39,46,47,30,42,45,48,31,24,37,40);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19,29,37,20,31,35,45,22,30,36,46,23,33,21,32)(18,24,38,43,34,42,26,40,27,39,44,48,25,28,41,47);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(19,30)(24,39)(28,42)(31,33)(32,45)(37,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^b=egi,d^c=dgi,e^c=egi,f^c=fgi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45020);