Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131804 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
581 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 581 conjugacy classes of subgroups — too many to draw. Only its 116 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:
(2,6)(3,7)(4,8)(9,10,12,18,13,29,26,40,15,20,28,37,14,19,27,36)(11,21,22,31,30,35,39,33,24,34,38,32,23,17,25,16);(1,2,5,3)(4,7,8,6)(9,11)(10,16)(12,25)(13,23)(14,30)(15,24)(17,18)(19,31)(20,33)(21,36)(22,27)(26,38)(28,39)(29,32)(34,40)(35,37);(1,3,5,2)(4,6,8,7)(9,11)(10,17)(12,22)(13,30)(14,23)(15,24)(16,18)(19,34)(20,35)(21,29)(25,27)(26,39)(28,38)(31,40)(32,36)(33,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=egi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39146);