Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×M16 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C22 | #1352 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,3,5,4,6,7,8)(9,10,13,18)(11,21,23,50)(12,17,28,41)(14,34,33,64)(15,19,32,47)(16,36,37,26)(20,48,46,27)(22,39,52,44)(24,57,56,72)(25,51,55,70)(29,43,61,45)(30,62,60,71)(31,63,35,40)(38,65,66,58)(42,68,67,59)(49,54,69,53);(9,11,14,35,15,25,27,24)(10,16,34,39,19,38,20,49)(12,26,30,22,29,58,59,53)(13,31,33,55,32,56,48,23)(17,40,62,51,43,57,42,21)(18,44,64,66,47,69,46,37)(28,54,60,36,61,52,68,65)(41,50,71,63,45,70,67,72)Polycyclic presentation:
2^9|a^2=d,b^2=fh,c^2=fi,d^2=g,f^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=def,f^b=fi,h^b=hi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1181);