Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #331927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #532 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10)(3,11,14,27)(4,15,18,28)(5,19,22,9)(7,20,23,13)(8,21,26,17)(12,30,31,24)(16,25,32,29)(33,34,39,44)(35,45,48,53)(36,42,51,57)(37,40,38,43)(41,52,56,46)(47,61,63,59)(49,58,50,54)(55,60,64,62);(1,3)(2,7)(4,16)(5,20)(6,14)(8,25)(9,27)(10,23)(11,19)(12,17)(13,22)(15,24)(18,32)(21,31)(26,29)(28,30)(34,40)(35,46)(36,49)(37,38)(42,57)(43,44)(45,53)(48,52)(50,51)(55,61)(59,64)(60,62);(7,23)(11,27)(24,30)(25,29)(33,35,39,48)(34,41,44,56)(36,47,51,63)(37,46,38,52)(40,53,43,45)(42,55,57,64)(49,62,50,60)(54,61,58,59);(1,4,6,18)(2,8,10,26)(3,12,14,31)(5,21,22,17)(7,24,23,30)(9,28,19,15)(11,29,27,25)(13,32,20,16)(33,36,39,51)(34,42,44,57)(35,47,48,63)(37,50,38,49)(40,54,43,58)(41,55,56,64)(45,59,53,61)(46,60,52,62)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 239142);