Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94303 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1393 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C2 | #7250 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 64 points:
(1,2)(3,13,15,31)(4,18)(5,22)(6,25)(7,12)(8,14,28,39)(9,20)(10,23)(11,27)(16,43,42,62)(17,46,26,38)(19,32)(21,49)(24,56)(29,59,58,64)(30,52,36,41)(33,53)(34,51)(35,47)(37,45,61,57)(40,54,44,60)(48,63)(50,55);(1,3,5,16,7,15,24,42)(2,8,10,29,12,28,35,58)(4,14,19,40,21,39,51,44)(6,26,23,57,27,17,47,45)(9,31,32,61,33,13,34,37)(11,36,22,54,25,30,56,60)(18,38,48,62,49,46,55,43)(20,52,50,64,53,41,63,59)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gh,h^b=hi,e^c=eh,g^c=gi,g^e=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1516);