Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1867 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 421 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,14,19,15,20,13,17)(11,21,23,36,24,38,27,37)(12,18,25,34,26,35,28,33)(16,29,31,40,32,39,30,22);(1,2,6,3)(4,8,5,7)(9,11,14,23,15,24,13,27)(10,16,19,31,20,32,17,30)(12,22,25,29,26,40,28,39)(18,21,34,36,35,38,33,37);(1,3,6,2)(4,8,5,7)(10,17,20,19)(16,30,32,31)(18,34,35,33)(21,36,38,37)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=gh,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,h^a=hi,d^b=dg,e^b=ehi,d^c=dh,e^c=eg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43458);