Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9221 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,6,12,8,11,29,41,5,13,15,44,7,30,28,32)(3,14,19,34,17,36,26,60,20,9,18,35,31,37,46,42)(4,21,23,39,24,57,47,40,25,58,59,38,22,43,27,10)(16,48,50,63,52,56,54,45,53,62,49,61,51,64,55,33);(1,3,15,46,8,31,28,18,5,20,6,26,7,17,29,19)(2,9,32,34,30,37,44,60,13,14,41,35,11,36,12,42)(4,16,47,50,22,52,23,54,25,53,27,49,24,51,59,55)(10,33,21,56,39,61,57,48,40,45,58,64,38,63,43,62);(1,4)(2,10)(3,16)(5,25)(6,27)(7,24)(8,22)(9,33)(11,38)(12,43)(13,40)(14,45)(15,47)(17,51)(18,54)(19,55)(20,53)(21,32)(23,28)(26,49)(29,59)(30,39)(31,52)(34,56)(35,64)(36,63)(37,61)(41,58)(42,62)(44,57)(46,50)(48,60)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45859);