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 | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 608 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)(9,10,13,19,14,20,29,37,15,17,28,34,12,18,26,35)(11,21,24,38,25,16,30,32,22,33,27,40,23,31,39,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,17,31)(12,23)(13,27)(14,25)(15,22)(18,21,20,33)(19,36,34,32)(24,28)(26,30)(29,39)(35,40,37,38);(3,8)(5,7)(10,17)(13,28)(16,31)(18,20)(21,33)(24,27)(26,29)(30,39);(2,6)(5,7)(11,22)(16,31)(21,33)(23,25)(24,27)(30,39)(32,36)(38,40)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,d^b=di,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403824);