Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94787 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7493 |
| commutator | C4⋊Q8 | C42 |
| Frattini, cosocle | #3152 | C22 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3152 ◁ G | |
| LECS | G ▷ #3152 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 34 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:
(1,2,5,10,7,13,28,43)(3,14,16,45,18,48,25,59)(4,19,21,56,23,44,58,40)(6,12,29,41,30,11,27,42)(8,31,32,51,35,52,63,50)(9,36,37,24,38,26,60,22)(15,33,49,39,20,34,57,62)(17,47,54,64,55,46,53,61);(1,3,6,17,7,18,30,55)(2,8,11,33,13,35,12,34)(4,20,22,51,23,15,24,50)(5,25,27,54,28,16,29,53)(9,14,19,46,38,48,44,47)(10,39,41,32,43,62,42,63)(21,49,26,52,58,57,36,31)(37,64,40,59,60,61,56,45)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfh,f^b=fh,g^b=ghi,d^c=dgh,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1986);