Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187667 |
| 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 | #62743 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2042 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)(4,23,27,51,29,52,48,41)(5,30,32,13,34,47,49,20)(7,37,35,18,39,54,33,11)(8,15,36,12,31,22,50,19)(10,40,43,62,17,24,42,26)(25,60,28,46,61,64,57,56)(38,44,63,45,58,53,59,55);(1,3)(2,9)(4,17)(5,18)(6,12)(7,37)(8,22)(10,29)(11,34)(13,32)(14,36)(15,31)(16,19)(20,49)(21,50)(23,38)(24,51)(25,42)(26,46)(27,45)(28,53)(30,35)(33,47)(39,54)(40,41)(43,61)(44,57)(48,55)(52,58)(56,62)(59,64)(60,63);(1,4)(2,10)(3,17)(5,24)(6,27)(7,38)(8,28)(9,29)(11,41)(12,45)(13,46)(14,43)(15,44)(16,48)(18,51)(19,55)(20,56)(21,42)(22,53)(23,37)(25,50)(26,32)(30,60)(31,57)(33,59)(34,40)(35,63)(36,61)(39,58)(47,64)(49,62)(52,54);(1,5,31,35)(2,11,22,47)(3,18,15,30)(4,24,57,63)(6,32,50,39)(7,16,49,36)(8,33,9,34)(10,41,53,64)(12,13,21,54)(14,37,19,20)(17,51,44,60)(23,55,56,43)(25,58,27,26)(28,59,29,40)(38,48,62,61)(42,52,45,46)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94933);