Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #236335 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,19)(14,25)(15,24)(17,23)(20,26)(21,27)(22,31)(28,30)(29,32)(33,34)(35,43)(36,40)(37,49)(38,42)(39,46)(41,52)(44,60)(45,58)(47,59)(48,57)(50,54)(51,56)(53,63)(55,62)(61,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,25)(10,13)(11,23)(12,24)(18,28)(19,29)(20,31)(21,30)(22,27)(26,32)(33,35,38,45)(34,39,42,54)(36,44,48,61)(37,50,52,46)(40,55,57,53)(41,58,49,43)(47,63,51,62)(56,64,59,60);(1,4,7,20)(2,9,12,27)(3,14,17,31)(5,19,6,21)(8,22,24,25)(10,26,11,18)(13,28,23,32)(15,29,16,30)(33,36,38,48)(34,40,42,57)(35,44,45,61)(37,51,52,47)(39,53,54,55)(41,59,49,56)(43,60,58,64)(46,63,50,62);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,13,24,23)(9,26,27,18)(14,29,31,30)(22,28,25,32)(35,45)(39,54)(43,58)(44,61)(46,50)(53,55)(60,64)(62,63)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143550);