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 | #45975 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10,15,14)(11,19,21,16)(12,17,20,22)(13,18,23,24)(26,31)(29,42)(32,36)(33,44)(38,51)(41,43)(46,48)(53,54);(1,2)(3,7)(4,6)(5,8)(9,11,15,21)(10,16,14,19)(12,13,20,23)(17,24,22,18)(26,31)(27,35)(28,39)(34,45)(38,51)(41,43)(46,48)(55,56);(1,3)(2,5)(4,8)(6,7)(9,12,15,20)(10,17,14,22)(11,13,21,23)(16,24,19,18)(25,26,30,31)(27,36,35,32)(28,33,39,44)(29,34,42,45)(37,48,40,46)(38,52,51,47)(41,49,43,50)(53,56,54,55);(2,6)(5,7)(9,11)(10,16)(12,13)(14,19)(15,21)(17,24)(18,22)(20,23)(25,27,30,35)(26,32,31,36)(28,40,39,37)(29,38,42,51)(33,48,44,46)(34,47,45,52)(41,54,43,53)(49,56,50,55);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,20)(12,21)(14,24)(15,23)(16,17)(19,22)(25,28,30,39)(26,33,31,44)(27,37,35,40)(29,43,42,41)(32,46,36,48)(34,50,45,49)(38,54,51,53)(47,56,52,55);(1,3)(2,7)(4,8)(5,6)(9,14,15,10)(11,19,21,16)(12,22,20,17)(13,18,23,24)(25,29)(26,34)(27,38)(28,41)(30,42)(31,45)(32,47)(33,49)(35,51)(36,52)(37,53)(39,43)(40,54)(44,50)(46,55)(48,56)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126703);