Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
592 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 178 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 — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,20,14,16,29,31,15,19,27,35,12,18,26,34)(11,21,24,36,25,32,38,40,22,17,28,33,23,30,37,39);(1,3)(2,5)(4,8)(6,7)(10,16,19,18)(13,27)(17,30,21,32)(20,34,35,31)(24,28)(26,29)(33,40,36,39)(37,38);(3,8)(5,7)(9,11)(10,17)(12,23)(13,28)(14,25)(15,22)(16,30)(18,32)(19,21)(20,36)(24,27)(26,38)(29,37)(31,40)(33,35)(34,39);(2,6)(5,7)(11,22)(17,21)(23,25)(24,28)(30,32)(33,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=ceh,g^b=gi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403823);