Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #459 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8947 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15,19,48,21,44,20,36)(4,22,25,40,27,58,26,45)(5,28)(6,12)(7,13)(8,14)(9,17,37,18,39,54,38,29)(10,24,41,59,43,30,42,23)(11,31)(16,49)(32,46)(33,34)(35,56)(47,55)(50,60)(51,57)(52,61)(53,62)(63,64);(1,3,6,17,32,20,5,29,8,21,33,54,7,19,31,18)(2,9,34,15,46,38,11,36,14,39,12,44,13,37,28,48)(4,16,23,51,26,56,30,50,27,53,59,64,25,52,24,55)(10,35,40,47,42,62,22,57,43,61,45,60,41,49,58,63);(1,4)(2,10)(3,16)(5,30)(6,23)(7,25)(8,27)(9,35)(11,22)(12,45)(13,41)(14,43)(15,47)(17,51)(18,55)(19,52)(20,56)(21,53)(24,31)(26,32)(28,58)(29,50)(33,59)(34,40)(36,57)(37,49)(38,62)(39,61)(42,46)(44,60)(48,63)(54,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=eg,f^c=fg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43877);