Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,3)(4,5)(7,8)(9,10,14,18,15,20,12,16)(11,21,22,38,25,36,29,37)(13,27,28,33,23,31,26,19)(17,32,30,24,35,39,34,40);(1,2,4,3,6,8,5,7)(10,16,20,18)(11,22,25,29)(13,23)(19,27,33,31)(21,36)(24,32,40,39)(26,28)(37,38);(1,3,4,8,6,7,5,2)(9,11,23,40,12,22,28,32,15,25,13,24,14,29,26,39)(10,17,33,36,18,34,31,38,20,35,19,21,16,30,27,37)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eg,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ch,e^b=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309605, #72309628, #72309641, … |
| maximal normal subgroup of | #72309605, #72309628, #72309641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48957);