Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,50)(4,21,27,58)(5,11,16,37)(6,12,33,32)(7,34,19,44)(9,36,39,18)(10,22,42,25)(13,30,47,31)(17,51,54,62)(23,61,40,60)(24,56,28,59)(26,48,35,64)(29,43,45,49)(38,52,57,63)(41,55,46,53);(1,3,16,36)(2,9,37,50)(4,22,28,61)(5,18,8,20)(6,31,45,7)(10,21,23,59)(11,15,14,39)(12,44,49,47)(13,32,34,43)(17,52,55,35)(19,33,30,29)(24,60,27,25)(26,54,63,53)(38,51,48,46)(40,56,42,58)(41,57,62,64);(1,4,5,28)(2,10,11,23)(3,17,18,55)(6,25,29,60)(7,35,30,52)(8,27,16,24)(9,38,15,48)(12,21,43,59)(13,46,44,51)(14,42,37,40)(19,26,31,63)(20,54,36,53)(22,45,61,33)(32,58,49,56)(34,62,47,41)(39,57,50,64);(1,2,5,11)(3,9,18,15)(4,23,28,10)(6,32,29,49)(7,13,30,44)(8,14,16,37)(12,45,43,33)(17,48,55,38)(19,47,31,34)(20,39,36,50)(21,22,59,61)(24,42,27,40)(25,56,60,58)(26,62,63,41)(35,51,52,46)(53,57,54,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=i,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161757);