Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2075 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11379 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,33,38,31,12,9,15,7,34,8,14,5,30)(3,16,19,10,36,40,18,50,22,52,20,42,21,48,32,41)(4,23,27,46,37,49,35,11,29,45,26,51,28,44,25,47)(17,53,55,62,59,39,54,43,57,64,24,61,56,60,58,63);(1,3,8,36,9,22,33,21)(2,10,38,41,15,42,14,50)(4,24,37,54,29,55,28,58)(5,19,7,18,31,20,6,32)(11,43,46,61,47,63,51,62)(12,48,34,52,30,40,13,16)(17,25,56,27,57,35,59,26)(23,60,44,53,45,39,49,64);(1,4,8,37,9,29,33,28)(2,11,38,46,15,47,14,51)(3,17,36,56,22,57,21,59)(5,27,7,35,31,26,6,25)(10,39,41,64,42,60,50,53)(12,23,34,44,30,45,13,49)(16,43,48,61,52,63,40,62)(18,24,20,54,32,55,19,58)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56610);