Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139571 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9469 |
| prime radical | #525 | D4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,13,17,40)(4,19,14,42)(5,23)(7,28,20,12)(8,29,31,15)(9,27)(10,35,37,22)(16,46)(18,49,43,41)(21,52)(24,38)(25,50)(26,34)(30,47)(32,45,39,58)(33,53)(36,57)(44,54)(48,60)(51,55)(56,63)(59,62)(61,64);(1,3,14,29,20,43,22,45,6,17,4,15,7,18,35,58)(2,8,10,13,28,39,42,49,11,31,37,40,12,32,19,41)(5,24,25,44,51,48,53,64,27,16,21,47,26,56,57,62)(9,30,36,38,55,61,52,63,23,54,33,46,34,59,50,60);(2,9)(8,30)(10,36)(11,23)(12,34)(13,38)(19,50)(28,55)(31,54)(32,59)(33,37)(39,61)(40,46)(41,60)(42,52)(49,63)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72320761, #72342959, #72342989, … |
| maximal normal subgroup of | #72320761, #72342959, #72342989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46773);