Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #357 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7154 |
| prime radical | C23 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,5,13,8,16,28,15,10,17,7,36,9,37,33,47)(3,18,20,49,23,50,51,39,25,43,22,58,24,59,55,60)(4,26,29,46,30,62,35,61,32,14,6,12,31,44,34,45)(11,21,38,53,41,56,40,27,42,52,48,54,63,57,64,19);(1,3,6,21,7,22,30,54,8,23,34,56,33,55,32,19,10,25,29,52,5,20,31,53,9,24,35,57,28,51,4,27)(2,11,14,39,15,40,45,59,37,63,62,49,13,38,12,43,17,42,26,60,47,64,61,50,16,41,44,58,36,48,46,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225305, #72225311, #72225312, … |
| maximal normal subgroup of | #72225305, #72225311, #72225312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1225);