Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140220 |
| 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 | |
219 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 219 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)(3,11,14,15,13,8,12,16)(17,18)(19,29)(20,32,34,40)(21,35)(22,24)(23,28)(25,30,43,48)(26,37)(27,31)(33,45,36,46)(38,44,47,42)(39,41);(2,7,10,9)(8,15,11,16)(17,19,21,22,23,31,37,39)(18,24,26,29,28,41,35,27)(20,30,36,38,34,48,33,47)(25,40,44,46,43,32,42,45);(1,3,4,12,6,13,5,14)(2,8,9,16,10,11,7,15)(17,20,23,34)(18,25,28,43)(19,30,31,48)(21,36,37,33)(22,38,39,47)(24,40,41,32)(26,44,35,42)(27,45,29,46)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,e^b=ei,d^c=di,e^c=egh,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72287105, #72287109, #72287188, … |
| maximal normal subgroup of | #72287105, #72287109, #72287188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47507);