Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C23 | #452 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
270 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 270 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,14,20,16,21,15,31)(11,17,23,34,24,19,13,29)(12,25,28,33,26,40,32,38)(18,30,36,22,35,39,37,27);(1,2)(3,4)(5,7)(6,8)(9,11,14,23,16,24,15,13)(10,17,20,34,21,19,31,29)(12,22,28,39,26,27,32,30)(18,33,36,40,35,38,37,25);(1,3,6,7)(2,4,8,5)(9,12)(10,18)(11,22)(13,30)(14,28)(15,32)(16,26)(17,33)(19,38)(20,36)(21,35)(23,39)(24,27)(25,29)(31,37)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=egi,d^c=dgi,e^c=eh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43722);