Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| 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 | |
392 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 392 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:
(1,2,5,8)(3,6,4,7)(9,10,13,17,14,18,20,29)(11,19,23,35,21,33,36,40)(12,24,27,37,26,16,32,34)(15,28,31,25,30,38,39,22);(2,6)(3,4)(7,8)(9,11,20,36,14,21,13,23)(10,15,29,39,18,30,17,31)(12,25,32,28,26,22,27,38)(16,33,37,35,24,19,34,40);(2,7)(3,4)(6,8)(11,21)(12,26)(15,30)(19,33)(23,36)(27,32)(31,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,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, 43411);