Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1347 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7120 |
| prime radical | C23 | #359 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,5,11,6,13,27,44,31,48,61,64,24,42,26,10,8,12,29,43,33,47,25,59,7,14,28,46,32,50,4,22)(3,15,17,51,19,53,16,49,34,63,55,37,30,62,60,38,21,45,58,40,18,41,57,52,20,54,23,9,35,36,56,39);(1,3)(2,9,12,37)(4,23)(5,16)(6,30)(7,34)(8,21)(10,41,22,53)(11,45,43,15)(13,49,47,52)(14,51,48,40)(17,25)(18,32)(19,24)(20,31)(26,55)(27,58)(28,60)(29,57)(33,35)(36,59,62,44)(38,42,39,50)(46,54,64,63)(56,61)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=cf,d^b=defi,f^b=fh,g^b=ghi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1132);