Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #326487 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2563 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
552 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 4 generators on 40 points:
(1,2,3,6,4,7,8,5)(9,10,14,20)(11,17)(12,24,26,30)(13,16,15,19)(18,27,34,25)(21,29)(22,36)(23,31)(28,33)(32,35)(37,39)(38,40);(10,16,20,19)(11,21,23,28)(17,31)(18,30,34,24)(22,35,37,38)(29,33)(32,40)(36,39);(2,5)(3,8)(6,7)(9,11)(10,17)(12,22)(13,21)(14,23)(15,28)(16,29)(18,32)(19,33)(20,31)(24,36)(25,35)(26,37)(27,38)(30,39)(34,40);(9,12,15,27,14,26,13,25)(10,18,16,30,20,34,19,24)(11,22,28,38,23,37,21,35)(17,32,29,39,31,40,33,36)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 233685);