Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147512 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10960 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
806 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 3 generators on 64 points:
(1,2,6,13,8,15,31,43)(3,16,19,46,21,34,55,36)(4,22,26,40,28,42,30,58)(5,12,23,14,29,11,7,32)(9,20,35,18,37,48,44,54)(10,38,39,24,41,27,56,25)(17,47,51,62,53,33,57,59)(45,52,60,50,61,63,64,49);(1,3,7,20,8,21,23,48)(2,9,14,36,15,37,32,46)(4,17,27,52,28,53,38,63)(5,18,6,19,29,54,31,55)(10,33,40,60,41,47,58,64)(11,34,13,44,12,16,43,35)(22,45,56,62,42,61,39,59)(24,50,30,51,25,49,26,57);(1,4,23,38,8,28,7,27)(2,10,14,40,15,41,32,58)(3,17,48,63,21,53,20,52)(5,24,31,26,29,25,6,30)(9,33,36,60,37,47,46,64)(11,42,13,39,12,22,43,56)(16,45,35,62,34,61,44,59)(18,50,55,57,54,49,19,51)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54711);