Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,7)(3,4,8,6)(9,10,15,11)(12,18,14,16)(13,17,22,19)(20,24,21,23)(25,26,31,29)(27,35,36,32)(28,39,42,33)(30,43,45,34)(37,46,40,48)(38,47,44,50)(41,49,54,52)(51,56,53,55);(1,3,5,8)(2,6,7,4)(9,11,15,10)(12,18,14,16)(13,17,22,19)(20,23,21,24)(27,36)(32,35)(37,40)(38,44)(46,48)(47,50)(51,53)(55,56);(1,2)(3,4)(5,7)(6,8)(9,12,15,14)(10,16,11,18)(13,20,22,21)(17,23,19,24)(25,27,31,36)(26,32,29,35)(28,40,42,37)(30,44,45,38)(33,48,39,46)(34,50,43,47)(41,51,54,53)(49,55,52,56);(1,4)(2,3)(5,6)(7,8)(25,28,31,42)(26,33,29,39)(27,37,36,40)(30,41,45,54)(32,46,35,48)(34,49,43,52)(38,51,44,53)(47,55,50,56);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,31,26)(27,35,36,32)(28,33,42,39)(30,43,45,34)(37,46,40,48)(38,50,44,47)(41,49,54,52)(51,55,53,56);(1,4)(2,3)(5,6)(7,8)(9,14,15,12)(10,18,11,16)(13,21,22,20)(17,24,19,23)(25,30,31,45)(26,34,29,43)(27,38,36,44)(28,41,42,54)(32,47,35,50)(33,49,39,52)(37,51,40,53)(46,55,48,56)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839745);