Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139428 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9428 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 332 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 — 3 generators on 64 points:
(1,2,7,13,8,12,31,11,6,14,32,46,33,15,5,28)(3,16,20,37,19,45,34,47,21,48,29,9,22,38,18,39)(4,23,26,49,27,43,24,59,25,44,30,41,35,10,40,42)(17,51,54,58,53,61,52,36,55,50,57,63,56,64,60,62);(1,3,8,22,6,21,33,19)(2,9,12,47,14,37,15,39)(4,17,35,53,25,55,27,56)(5,29,32,18,31,20,7,34)(10,36,44,63,43,62,23,58)(11,45,13,48,28,38,46,16)(24,54,30,60,40,57,26,52)(41,50,42,61,49,51,59,64);(1,4)(2,10)(3,17)(5,30)(6,25)(7,24)(8,35)(9,36)(11,42)(12,44)(13,49)(14,43)(15,23)(16,50)(18,57)(19,56)(20,52)(21,55)(22,53)(26,31)(27,33)(28,59)(29,60)(32,40)(34,54)(37,62)(38,64)(39,58)(41,46)(45,61)(47,63)(48,51)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gh,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eg,f^c=fgh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46624);