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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10)(11,19)(12,17)(13,24)(14,20)(15,21)(16,23)(18,22)(26,27)(28,38)(30,44)(33,36)(39,41)(42,45)(47,51)(55,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,23)(17,24)(19,21)(25,26,31,27)(28,35,38,32)(29,36,43,33)(30,34,44,37)(39,46,41,48)(40,51,52,47)(42,50,45,49)(53,55,54,56);(1,3)(2,5)(4,8)(6,7)(9,12,15,20)(10,17,21,14)(11,22,23,13)(16,24,19,18)(25,27,31,26)(28,32,38,35)(29,36,43,33)(30,37,44,34)(39,46,41,48)(40,47,52,51)(42,50,45,49)(53,55,54,56);(2,6)(5,7)(9,13)(10,18)(11,12)(14,19)(15,22)(16,17)(20,23)(21,24)(25,28,31,38)(26,32,27,35)(29,41,43,39)(30,40,44,52)(33,48,36,46)(34,47,37,51)(42,54,45,53)(49,56,50,55);(1,2)(3,5)(4,6)(7,8)(9,11)(10,19)(12,22)(13,20)(14,24)(15,23)(16,21)(17,18)(25,29)(26,33)(27,36)(28,39)(30,45)(31,43)(32,46)(34,50)(35,48)(37,49)(38,41)(40,54)(42,44)(47,56)(51,55)(52,53);(1,3)(2,7)(4,8)(5,6)(9,14,15,17)(10,20,21,12)(11,18,23,24)(13,19,22,16)(25,30)(26,34)(27,37)(28,40)(29,42)(31,44)(32,47)(33,49)(35,51)(36,50)(38,52)(39,53)(41,54)(43,45)(46,55)(48,56)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,f^2=gh,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, 10127571);