Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187177 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62826 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,27,34,49,60)(3,13,17,41,16,48,25,57)(4,19,22,52,21,37,54,62)(6,9,26,33,7,12,28,40)(8,29,32,15,45,47,39,46)(10,35,38,24,51,56,64,23)(14,42,44,63,18,36,50,61)(20,30,53,58,43,31,55,59);(1,3)(2,8)(4,20)(5,25)(6,14)(7,18)(9,30)(10,36)(11,39)(12,31)(13,37)(15,24)(16,27)(17,49)(19,48)(21,43)(22,55)(23,46)(26,50)(28,44)(29,56)(32,60)(33,59)(34,45)(35,47)(38,63)(40,58)(41,52)(42,51)(53,54)(57,62)(61,64);(2,9)(4,21)(5,26)(8,30)(10,37)(12,34)(13,36)(15,46)(17,44)(19,51)(20,43)(22,56)(23,24)(25,50)(28,49)(29,55)(31,45)(35,54)(38,64)(41,57)(42,48)(47,53)(52,62)(61,63);(2,9)(3,14)(10,19)(11,33)(12,34)(15,43)(16,18)(17,44)(20,46)(25,50)(29,53)(37,51)(38,52)(40,60)(47,55)(62,64)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94556);