Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #102 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7167 |
| prime radical | C23 | #350 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,12,8,11,6,31,7,15,29,46,28,43,32,13,9,16,30,42,34,44,33,47,26,14,49,45,4,24,27,48)(3,17,19,50,22,38,20,57,21,52,56,64,18,10,35,36,23,39,54,60,55,40,58,62,53,51,61,37,25,41,59,63);(1,3,6,20,28,18,30,54,26,53,27,59,8,22,29,56,9,23,33,58,4,25,5,19,7,21,32,35,34,55,49,61)(2,10,13,37,24,52,46,62,14,38,31,60,44,17,12,36,16,41,48,64,43,51,45,57,15,40,47,50,11,39,42,63)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fg,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225516, #72225517, #72225519 |
| maximal normal subgroup of | #72225516, #72225517, #72225519 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1238);