Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62752 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,9,3,16,21)(4,23,27,51,29,52,59,41)(5,30,32,45,34,46,49,54)(7,13,36,18,39,20,33,11)(8,15,37,12,31,22,50,19)(10,40,43,64,17,24,42,38)(25,61,28,47,35,63,57,55)(26,44,62,48,58,53,60,56);(1,3)(2,9)(4,17)(5,18)(6,12)(7,13)(8,22)(10,29)(11,34)(14,37)(15,31)(16,19)(20,39)(21,50)(23,26)(24,51)(25,42)(27,48)(28,53)(30,36)(32,45)(33,46)(35,43)(38,47)(40,41)(44,57)(49,54)(52,58)(55,64)(56,59)(60,63)(61,62);(1,4,8,28)(2,10,15,44)(3,17,22,53)(5,24,33,60)(6,35,37,27)(7,38,32,26)(9,29,31,57)(11,41,30,61)(12,43,14,48)(13,47,45,23)(16,25,50,59)(18,51,46,63)(19,42,21,56)(20,55,54,52)(34,40,36,62)(39,64,49,58);(1,5,31,36)(2,11,22,46)(3,18,15,30)(4,24,57,62)(6,32,50,39)(7,16,49,37)(8,33,9,34)(10,41,53,63)(12,45,21,20)(13,19,54,14)(17,51,44,61)(23,56,55,43)(25,58,27,38)(26,59,64,35)(28,60,29,40)(42,52,48,47)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94717);