Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #466 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9657 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,6,14,8,15,33,44,9,16,32,12,7,34,5,13)(3,17,21,46,23,47,31,10,24,37,20,38,22,36,19,39)(4,25,27,11,29,41,59,42,30,56,58,43,28,40,26,55)(18,35,50,60,52,61,57,54,53,62,49,45,51,63,48,64);(1,3,7,22,9,24,8,23)(2,10)(4,18,28,51,30,53,29,52)(5,31,32,21,33,19,6,20)(11,35)(12,37)(13,36)(14,17)(15,38)(16,39)(25,54)(26,57,58,50,59,48,27,49)(34,46)(40,60)(41,45)(42,61)(43,62)(44,47)(55,63)(56,64);(1,4,5,26,7,28,32,58,9,30,33,59,8,29,6,27)(2,11,14,41,15,42,44,56,16,43,12,40,34,55,13,25)(3,18,31,57,22,51,21,50,24,53,19,48,23,52,20,49)(10,35,17,45,38,61,47,64,39,62,37,60,46,63,36,54)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=di,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320349, #72320403, #72320408, … |
| maximal normal subgroup of | #72320349, #72320403, #72320408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47929);