Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2535 |
| prime radical | #2397 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
633 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 633 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(1,2,4,7)(3,5,8,6)(9,10)(11,19)(12,23)(13,26)(14,18)(15,21)(16,24)(17,27)(20,33)(22,31)(25,37)(28,36)(29,35)(30,38)(32,34)(39,40);(9,11)(10,15,18,28)(12,20)(13,21)(14,22)(16,29,30,33)(17,19,26,31)(23,32,37,39)(24,34)(25,35)(27,36)(38,40);(2,5)(3,8)(6,7)(15,28)(19,31)(29,33)(32,39);(9,12,13,24,14,25,27,38)(10,16,17,23,18,30,26,37)(11,20,21,34,22,35,36,40)(15,29,19,32,28,33,31,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307172, #72307197, #72307201, … |
| maximal normal subgroup of | #72307172, #72307197, #72307201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 224284);