Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12090 |
| 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 | |
433 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 433 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:
(2,6)(3,4)(7,8)(9,10,13,17,14,18,27,34)(11,19,21,36,22,32,37,40)(12,23,26,35,25,16,31,33)(15,28,30,24,29,38,39,20);(1,2)(3,8)(4,6)(5,7)(9,11,13,21,14,22,27,37)(10,15,17,30,18,29,34,39)(12,24,26,38,25,20,31,28)(16,32,33,40,23,19,35,36);(1,2)(3,6)(4,8)(5,7)(12,25)(20,24)(26,31)(28,38)Polycyclic presentation:
2^9|a^2=f,b^2=f,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=egi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43330);