Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94479 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7323 |
| commutator | #117 | C42 |
| Frattini, cosocle | #2960 | C22 |
| prime radical | C23 | #344 |
| LCS | G ▷ #117 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2960 ◁ G | |
| LECS | G ▷ #2960 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 40 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:
(1,2,5,11,7,12,28,44,8,14,29,46,6,13,27,45)(3,15,17,47,30,59,57,64,19,48,53,61,18,38,25,56)(4,20,22,54,31,10,37,39,23,40,26,58,24,41,55,43)(9,32,33,50,35,52,42,51,36,60,63,16,34,49,62,21);(1,3)(2,9,12,34,14,36,13,35)(4,21,23,51)(5,25)(6,30)(7,18)(8,19)(10,38,40,48,41,59,20,15)(11,42,44,33,46,62,45,63)(16,24,50,31)(17,27)(22,49,26,52)(28,53)(29,57)(32,37,60,55)(39,47,58,56,43,61,54,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=egh,h^a=hi,c^b=cg,d^b=dfh,f^b=fi,g^b=gh,d^c=di,e^c=ehi,g^c=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1677);