Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12463 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1675 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9215 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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:
(1,2,7,13,8,14,29,12,5,15,16,42,30,11,6,31)(3,9,20,34,18,35,47,33,22,36,55,40,21,45,19,43)(4,23,26,39,24,41,48,46,28,44,57,10,27,37,25,38)(17,49,52,61,53,56,64,58,50,62,63,32,54,59,51,60);(1,3,16,47,8,21,6,20,5,22,7,19,30,18,29,55)(2,9,31,34,11,35,42,33,15,36,12,40,14,45,13,43)(4,17,48,63,27,53,26,51,28,50,25,52,24,54,57,64)(10,32,37,62,38,58,23,56,39,61,41,49,46,60,44,59);(1,4)(2,10)(3,17)(5,28)(6,26)(7,25)(8,27)(9,32)(11,38)(12,41)(13,44)(14,46)(15,39)(16,48)(18,54)(19,52)(20,51)(21,53)(22,50)(23,42)(24,30)(29,57)(31,37)(33,56)(34,62)(35,58)(36,61)(40,49)(43,59)(45,60)(47,63)(55,64)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=ghi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,d^c=dh,e^c=eh,f^c=fhi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45766);