Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94676 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1412 |
| commutator | #376 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| socle | C2 | #7412 |
| prime radical | #532 | D4 |
| LCS | G ▷ #376 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 64 points:
(1,2,5,10)(3,14,16,43)(4,19,20,53)(6,11,27,39)(7,12,25,40)(8,29,30,58)(9,24,35,22)(13,26,42,28)(15,31,45,37)(17,34,23,57)(18,51,52,60)(21,41,54,38)(32,48,61,46)(33,59,62,47)(36,55,63,56)(44,49,64,50);(1,3,6,18,7,17,28,50)(2,8,13,33,12,32,11,31)(4,15,22,46,21,47,56,58)(5,23,26,52,25,16,27,49)(9,34,19,51,36,14,41,44)(10,37,39,61,40,62,42,30)(20,45,55,29,54,59,24,48)(35,60,38,57,63,64,53,43)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fh,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=def,e^b=ehi,d^c=dghi,e^c=ei,e^d=ehi,f^d=fi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1872);