Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7481 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1390 |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C2 | #7187 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,11,7,13,25,43,8,12,26,41,6,27,30,42)(3,14,16,44,17,46,51,64,18,45,52,59,28,53,23,57)(4,19,20,54,21,10,36,37,22,39,24,58,29,38,56,63)(9,31,32,15,35,47,40,49,33,55,60,50,34,61,62,48);(1,3)(2,9,12,33)(4,15)(5,23)(6,28)(7,17)(8,18)(10,14,38,45)(11,40,41,62)(13,35,27,34)(16,25)(19,53,39,46)(20,55)(21,49)(22,50)(24,31)(26,51)(29,48)(30,52)(32,43,60,42)(36,61)(37,57,63,64)(44,54,59,58)(47,56)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=df,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1288);