Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94229 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7236 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | #2931 | C22 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×M16 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,15,17,50)(4,21,23,46)(6,30,25,42)(7,32,35,41)(8,14,28,44)(9,36,37,54)(10,27,12,24)(13,29,49,22)(16,47,53,43)(18,57,55,40)(19,52,56,61)(20,51,59,64)(26,38,33,62)(31,45,34,48)(39,60,63,58);(1,3,6,18,22,20,27,36,8,19,31,53,7,33,23,60)(2,9,12,38,32,40,30,61,14,39,46,64,13,47,45,50)(4,16,5,26,25,58,29,17,24,55,28,59,34,54,35,56)(10,15,41,37,42,62,44,57,21,52,49,63,48,51,11,43)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,e^2=fgi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=ghi,h^a=hi,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1426);