Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188759 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#383 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 250 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:
(9,10)(11,20)(12,24,26,31)(13,18)(14,19)(15,23)(16,25,32,38)(17,27)(21,28,34,39)(22,30)(29,36)(33,35,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15)(12,21,26,34)(13,23,27,36)(16,28)(17,20)(18,30)(19,29)(24,37)(25,35,38,40)(31,33)(32,39);(2,5)(6,8)(9,12,13,25,14,26,27,38)(10,16,17,31,19,32,18,24)(11,21,23,35,22,34,36,40)(15,28,20,33,29,39,30,37);(1,3,4,7)(2,6,5,8)(11,22)(15,29)(20,30)(21,34)(23,36)(28,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266165);