Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| 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 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,10,14,18,15,20,30,19)(11,16,22,32,23,33,13,28)(12,24,25,31,27,39,26,40)(17,34,35,38,36,29,37,21);(1,2)(3,4)(5,7)(6,8)(9,11,14,22,15,23,30,13)(10,16,18,32,20,33,19,28)(12,21,25,34,27,38,26,29)(17,31,35,39,36,40,37,24);(1,3,6,7)(2,4,8,5)(9,12)(10,17)(11,21)(13,29)(14,25)(15,27)(16,31)(18,35)(19,37)(20,36)(22,34)(23,38)(24,28)(26,30)(32,39)(33,40)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=eg,d^c=dg,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43719);