Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133293 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
800 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(5,8)(9,10,14,20,15,19,29,36)(11,21,23,37,24,34,28,39)(12,18,25,35,13,17,27,33)(16,30,32,26,31,38,40,22);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,26)(14,28)(15,24)(17,21)(18,34)(19,31)(20,32)(23,29)(25,38)(27,30)(33,37)(35,39)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,13)(10,17,19,18)(11,22,24,26)(14,27,29,25)(16,21,31,34)(20,35,36,33)(23,38,28,30)(32,39,40,37)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40518);