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 | |
236 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 236 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)(11,21,23,47)(12,17,27,40)(14,20,32,45)(15,19,31,46)(16,34,35,55)(22,36,50,43)(24,48,53,70)(25,38,30,59)(26,37,54,63)(28,42,57,44)(29,41,56,68)(33,39,61,67)(49,65,60,72)(51,64,52,69)(58,66,62,71);(9,11)(10,16)(12,26,28,55)(13,30,31,60)(14,25)(15,24)(17,38,42,65)(18,43,46,69)(19,37)(20,36)(21,41,48,66)(22,29,51,58)(23,32,53,61)(27,50)(33,49)(34,62)(35,45,63,67)(39,64)(40,47)(44,70)(52,57)(54,56)(59,68)(71,72)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defhi,e^b=ei,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, 1159);