Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| 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 | C23 | #408 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 295 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)(3,11,13,15)(4,14,7,8)(5,9)(6,10)(12,16)(17,18,23,27,22,29,39,46,21,28,30,31,38,26,44,45)(19,20,33,36,32,37,43,48,34,24,40,41,47,35,25,42);(1,3,5,14,6,13,16,8)(2,7,9,15,10,4,12,11)(17,19,30,33,22,32,44,43,21,34,23,40,38,47,39,25)(18,24,27,36,29,35,46,48,28,20,31,41,26,37,45,42);(1,4,6,7)(2,8,10,14)(3,12,13,9)(5,11,16,15)(17,20,21,24)(18,25,28,43)(19,31,34,27)(22,37,38,35)(23,36,30,41)(26,40,29,33)(32,45,47,46)(39,48,44,42)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=ei,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43981);