Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9215 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| 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 | |
307 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 307 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:
(1,2,6,11,7,12,29,41,5,13,14,42,26,10,38,30)(3,8,18,33,16,34,45,64,20,35,17,32,19,43,27,39)(4,21,23,36,22,56,46,37,25,57,58,44,24,40,28,9)(15,47,50,59,51,55,53,31,48,60,49,61,52,63,54,62);(1,3,14,45,7,19,38,18,5,20,6,27,26,16,29,17)(2,8,30,33,10,34,42,64,13,35,41,32,12,43,11,39)(4,15,46,49,24,51,23,54,25,48,28,50,22,52,58,53)(9,31,21,55,36,59,56,47,37,62,57,63,44,61,40,60);(1,4)(2,9)(3,15)(5,25)(6,28)(7,24)(8,31)(10,36)(11,40)(12,44)(13,37)(14,46)(16,52)(17,53)(18,54)(19,51)(20,48)(21,30)(22,26)(23,38)(27,50)(29,58)(32,63)(33,55)(34,59)(35,62)(39,60)(41,57)(42,56)(43,61)(45,49)(47,64)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=ghi,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=eh,f^b=fhi,h^b=hi,d^c=dh,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, 45764);