Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134940 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1802 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18)(19,27,28,43)(20,23)(21,33)(22,36)(24,29,38,46)(25,35)(26,32)(30,41,37,45)(31,44)(34,42,48,40)(39,47);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(18,23)(24,38)(25,39)(26,36)(27,43)(33,44)(40,42)(41,45);(17,19)(18,24)(20,28)(21,34)(22,37)(23,38)(25,41)(26,42)(27,44)(29,47)(30,32)(31,48)(33,43)(35,46)(36,40)(39,45)Polycyclic presentation:
2^9|a^2=f,b^2=di,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^c=ehi,f^c=fi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287582, #72287597, #72287608, … |
| maximal normal subgroup of | #72287582, #72287597, #72287608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42246);