Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,11,18)(3,13,14,43)(4,17)(5,23)(9,32,36,64)(10,38,41,58)(12,31,29,49)(15,46,19,44)(16,25,45,60)(21,54)(22,50)(24,52,59,30)(26,27)(28,40,47,42)(33,63,37,62)(34,57,51,35)(39,56)(48,55,53,61);(1,2,4,18,20,29,50,31)(3,9,19,51,45,36,48,34)(5,24,21,38,23,41,56,30)(6,12,17,49,7,11,22,8)(10,39,52,27,59,54,58,26)(13,42,46,64,60,40,61,32)(14,33,15,47,16,37,53,28)(25,57,55,63,43,35,44,62);(1,3)(2,9)(4,19)(5,25)(6,14)(7,16)(8,28)(10,40)(11,37)(12,33)(13,27)(15,17)(18,51)(20,45)(21,55)(22,53)(23,43)(24,57)(26,60)(29,36)(30,62)(31,34)(32,52)(35,41)(38,63)(39,61)(42,59)(44,56)(46,54)(47,49)(48,50)(58,64);(3,14)(9,33)(13,43)(15,19)(16,45)(25,60)(28,34)(32,63)(35,42)(36,37)(40,57)(44,46)(47,51)(48,53)(55,61)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107324);