Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1867 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,4)(7,8)(9,10,14,19,15,20,13,17)(11,21,24,39,22,36,29,38)(12,25,27,35,28,18,30,37)(16,31,32,26,33,40,34,23);(1,2)(3,8)(4,6)(5,7)(9,11,14,24,15,22,13,29)(10,16,19,32,20,33,17,34)(12,26,27,40,28,23,30,31)(18,36,37,38,25,21,35,39);(1,2)(3,6)(4,8)(5,7)(10,17,20,19)(11,22)(16,32,33,34)(18,37,25,35)(21,38,36,39)(23,26)(24,29)(31,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=egi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43365);