Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140270 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1926 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #482 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
213 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,5,9,6,10,4,7)(3,11,14,15,12,8,13,16)(17,18)(19,29)(20,32,35,41)(21,36)(22,24)(23,28)(25,30,44,33)(26,38)(27,31)(34,46,37,47)(39,45,48,43)(40,42);(2,7,10,9)(3,12)(8,16,11,15)(13,14)(17,19,21,22,23,31,38,40)(18,24,26,29,28,42,36,27)(20,33,37,48,35,30,34,39)(25,32,45,46,44,41,43,47);(1,3,4,13,6,12,5,14)(2,8,9,16,10,11,7,15)(17,20,23,35)(18,25,28,44)(19,30,31,33)(21,37,38,34)(22,39,40,48)(24,41,42,32)(26,45,36,43)(27,46,29,47)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dh,d^2=g,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ei,d^c=di,e^c=egh,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72287414, #72287416, #72287440, … |
| maximal normal subgroup of | #72287414, #72287416, #72287440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47517);