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 | #7171 |
| prime radical | C23 | #359 |
| 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 | |
153 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 153 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:
(1,2,5,13,8,15,27,14,10,16,7,36,9,37,33,46)(3,17,19,48,22,50,51,49,24,40,21,57,23,58,55,59)(4,25,28,45,30,61,35,60,31,62,29,12,6,34,32,44)(11,38,39,20,42,54,41,26,43,52,47,53,63,56,64,18);(1,3,6,20,33,55,29,38,9,23,31,18,7,21,35,56,10,24,30,53,27,51,28,52,8,22,4,26,5,19,32,54)(2,11,12,40,36,47,61,48,15,42,44,58,46,64,62,49,16,43,45,17,13,39,34,57,37,63,60,50,14,41,25,59)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,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 | #72225304, #72225311, #72225314, … |
| maximal normal subgroup of | #72225304, #72225311, #72225314, … |
| maximal quotient of | #72227011 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1250);